L(s) = 1 | + 2·4-s + 26·9-s + 4·13-s + 3·16-s − 8·25-s − 48·29-s + 52·36-s − 36·41-s − 22·49-s + 8·52-s + 12·64-s + 8·73-s + 351·81-s − 16·100-s − 24·101-s − 96·116-s + 104·117-s − 86·121-s + 127-s + 131-s + 137-s + 139-s + 78·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | + 4-s + 26/3·9-s + 1.10·13-s + 3/4·16-s − 8/5·25-s − 8.91·29-s + 26/3·36-s − 5.62·41-s − 3.14·49-s + 1.10·52-s + 3/2·64-s + 0.936·73-s + 39·81-s − 8/5·100-s − 2.38·101-s − 8.91·116-s + 9.61·117-s − 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 13/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8505662985\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8505662985\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - T^{2} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + T^{2} )^{8} \) |
| 23 | \( ( 1 - 16 T^{2} - 66 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
good | 3 | \( ( 1 - 13 T^{2} + 26 p T^{4} - 304 T^{6} + 961 T^{8} - 304 p^{2} T^{10} + 26 p^{5} T^{12} - 13 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 7 | \( ( 1 + 11 T^{2} + 103 T^{4} + 605 T^{6} + 4036 T^{8} + 605 p^{2} T^{10} + 103 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 43 T^{2} + 1039 T^{4} + 17269 T^{6} + 216580 T^{8} + 17269 p^{2} T^{10} + 1039 p^{4} T^{12} + 43 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - T + 28 T^{2} - 64 T^{3} + 421 T^{4} - 64 p T^{5} + 28 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 37 T^{2} + 883 T^{4} - 21031 T^{6} + 412864 T^{8} - 21031 p^{2} T^{10} + 883 p^{4} T^{12} - 37 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + 107 T^{2} + 5599 T^{4} + 184709 T^{6} + 4186564 T^{8} + 184709 p^{2} T^{10} + 5599 p^{4} T^{12} + 107 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 + 12 T + 161 T^{2} + 1098 T^{3} + 7668 T^{4} + 1098 p T^{5} + 161 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 41 T^{2} + 3298 T^{4} - 100808 T^{6} + 4632121 T^{8} - 100808 p^{2} T^{10} + 3298 p^{4} T^{12} - 41 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 - 188 T^{2} + 16840 T^{4} - 966980 T^{6} + 40792462 T^{8} - 966980 p^{2} T^{10} + 16840 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 + 9 T + 134 T^{2} + 720 T^{3} + 6945 T^{4} + 720 p T^{5} + 134 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 + 164 T^{2} + 13000 T^{4} + 689564 T^{6} + 30515854 T^{8} + 689564 p^{2} T^{10} + 13000 p^{4} T^{12} + 164 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - 14 T^{2} + 1225 T^{4} - 83114 T^{6} - 4288436 T^{8} - 83114 p^{2} T^{10} + 1225 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 32 T^{2} + 4686 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 - 160 T^{2} + 12174 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 245 T^{2} + 37255 T^{4} - 3637451 T^{6} + 263136244 T^{8} - 3637451 p^{2} T^{10} + 37255 p^{4} T^{12} - 245 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 + 248 T^{2} + 36160 T^{4} + 3690488 T^{6} + 285475582 T^{8} + 3690488 p^{2} T^{10} + 36160 p^{4} T^{12} + 248 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 - 353 T^{2} + 58498 T^{4} - 6303392 T^{6} + 507173521 T^{8} - 6303392 p^{2} T^{10} + 58498 p^{4} T^{12} - 353 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 73 | \( ( 1 - 2 T + 169 T^{2} + 280 T^{3} + 13108 T^{4} + 280 p T^{5} + 169 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 79 | \( ( 1 + 308 T^{2} + 43528 T^{4} + 3818540 T^{6} + 289603822 T^{8} + 3818540 p^{2} T^{10} + 43528 p^{4} T^{12} + 308 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 + 80 T^{2} + 4686 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 316 T^{2} + 56872 T^{4} - 7159588 T^{6} + 706979950 T^{8} - 7159588 p^{2} T^{10} + 56872 p^{4} T^{12} - 316 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 245 T^{2} + 43159 T^{4} - 5861171 T^{6} + 624798436 T^{8} - 5861171 p^{2} T^{10} + 43159 p^{4} T^{12} - 245 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.03154882026283112225820014468, −2.93793721770732683501032832126, −2.93287693108124007169243158511, −2.79589200665426229532317916033, −2.68621078566730402084723493977, −2.40846990631220874342852280966, −2.36484423161520116060606122481, −2.32728793589835670980954975762, −2.23552376405808972608003418169, −1.98872000927601495813380200073, −1.92068470097365812661544365976, −1.91647240613838261875622369087, −1.88627315307182738977593327741, −1.75842431750839318442323475547, −1.72019675183603140335098774017, −1.53902113191097589582746498338, −1.47852036871197510685108011312, −1.43954507740041303317078952382, −1.43856864676911672686469167644, −1.43662845010846257209249587670, −1.22206993851212959884778812761, −0.935773567118324688943439924890, −0.76397020552538127452632406971, −0.40103135245474932869507480706, −0.05531188581005735046006962523,
0.05531188581005735046006962523, 0.40103135245474932869507480706, 0.76397020552538127452632406971, 0.935773567118324688943439924890, 1.22206993851212959884778812761, 1.43662845010846257209249587670, 1.43856864676911672686469167644, 1.43954507740041303317078952382, 1.47852036871197510685108011312, 1.53902113191097589582746498338, 1.72019675183603140335098774017, 1.75842431750839318442323475547, 1.88627315307182738977593327741, 1.91647240613838261875622369087, 1.92068470097365812661544365976, 1.98872000927601495813380200073, 2.23552376405808972608003418169, 2.32728793589835670980954975762, 2.36484423161520116060606122481, 2.40846990631220874342852280966, 2.68621078566730402084723493977, 2.79589200665426229532317916033, 2.93287693108124007169243158511, 2.93793721770732683501032832126, 3.03154882026283112225820014468
Plot not available for L-functions of degree greater than 10.