| L(s) = 1 | + 24·11-s − 14·16-s + 36·41-s − 4·61-s − 81-s − 72·101-s + 220·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 336·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 7.23·11-s − 7/2·16-s + 5.62·41-s − 0.512·61-s − 1/9·81-s − 7.16·101-s + 20·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s − 25.3·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.652539219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.652539219\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{4} \) |
| good | 2 | \( ( 1 + 7 T^{4} + 33 T^{8} + 7 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 3 | \( 1 + T^{4} + 40 p T^{8} - 281 T^{12} + 7519 T^{16} - 281 p^{4} T^{20} + 40 p^{9} T^{24} + p^{12} T^{28} + p^{16} T^{32} \) |
| 7 | \( ( 1 + 23 T^{4} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{8} \) |
| 13 | \( 1 + 161 T^{4} + 25600 T^{8} - 9144961 T^{12} - 1555957361 T^{16} - 9144961 p^{4} T^{20} + 25600 p^{8} T^{24} + 161 p^{12} T^{28} + p^{16} T^{32} \) |
| 17 | \( 1 - 151 T^{4} + 73720 T^{8} + 32912111 T^{12} - 6287375201 T^{16} + 32912111 p^{4} T^{20} + 73720 p^{8} T^{24} - 151 p^{12} T^{28} + p^{16} T^{32} \) |
| 23 | \( ( 1 + 697 T^{4} + 205968 T^{8} + 697 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 95 T^{2} + 5188 T^{4} - 204725 T^{6} + 6425263 T^{8} - 204725 p^{2} T^{10} + 5188 p^{4} T^{12} - 95 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 - 19 T^{2} - 519 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 1436 T^{4} + 1671846 T^{8} - 1436 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 - 9 T + 112 T^{2} - 765 T^{3} + 6651 T^{4} - 765 p T^{5} + 112 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 43 | \( 1 + 1433 T^{4} + 2248576 T^{8} - 10077843337 T^{12} - 18670611586721 T^{16} - 10077843337 p^{4} T^{20} + 2248576 p^{8} T^{24} + 1433 p^{12} T^{28} + p^{16} T^{32} \) |
| 47 | \( ( 1 - 1823 T^{4} - 1556352 T^{8} - 1823 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 53 | \( 1 - 1774 T^{4} - 12772655 T^{8} - 246176206 T^{12} + 166346201317444 T^{16} - 246176206 p^{4} T^{20} - 12772655 p^{8} T^{24} - 1774 p^{12} T^{28} + p^{16} T^{32} \) |
| 59 | \( ( 1 + 10 T^{2} - 407 T^{4} - 64550 T^{6} - 12208412 T^{8} - 64550 p^{2} T^{10} - 407 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + T - 20 T^{2} - 101 T^{3} - 3341 T^{4} - 101 p T^{5} - 20 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 - 626 T^{4} - 19759245 T^{8} - 626 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 + 127 T^{2} + 11088 T^{4} + 127 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( 1 - 7 T^{4} + 56325016 T^{8} + 791850143 T^{12} + 2366040424143199 T^{16} + 791850143 p^{4} T^{20} + 56325016 p^{8} T^{24} - 7 p^{12} T^{28} + p^{16} T^{32} \) |
| 79 | \( ( 1 - 127 T^{2} + 7816 T^{4} + 529463 T^{6} - 70505201 T^{8} + 529463 p^{2} T^{10} + 7816 p^{4} T^{12} - 127 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 - 22409 T^{4} + 220029681 T^{8} - 22409 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 - 182 T^{2} + 15481 T^{4} - 327782 T^{6} - 13841996 T^{8} - 327782 p^{2} T^{10} + 15481 p^{4} T^{12} - 182 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( 1 + 19433 T^{4} + 136026016 T^{8} + 1254534451463 T^{12} + 17108273701697359 T^{16} + 1254534451463 p^{4} T^{20} + 136026016 p^{8} T^{24} + 19433 p^{12} T^{28} + p^{16} T^{32} \) |
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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
−3.03786233257994063421817908794, −3.03203500587748826523610814656, −2.86034621660371860889836957176, −2.79449494664242848019303923963, −2.60738705040407875798683767262, −2.60110875351930158529905306754, −2.44828283964513632151207691604, −2.31878033433103298259082609196, −2.26938920693951858837458204791, −2.21877903662779709649506577429, −2.04064615069188597893488373425, −2.02297638377880217074379395774, −2.00105872890907254017453475198, −1.60037486686541407068355549531, −1.59941447386744975880823963972, −1.59782972851092641143125665587, −1.58258844511942871919606405898, −1.31405509403724276617299966708, −1.17477598950885987912058227078, −1.12111598249255504373975236066, −1.06159829095848539463360883160, −0.932561809069326698935291249488, −0.67587766100900055015764193909, −0.53801773283226405451169975630, −0.16736495516484459892608290966,
0.16736495516484459892608290966, 0.53801773283226405451169975630, 0.67587766100900055015764193909, 0.932561809069326698935291249488, 1.06159829095848539463360883160, 1.12111598249255504373975236066, 1.17477598950885987912058227078, 1.31405509403724276617299966708, 1.58258844511942871919606405898, 1.59782972851092641143125665587, 1.59941447386744975880823963972, 1.60037486686541407068355549531, 2.00105872890907254017453475198, 2.02297638377880217074379395774, 2.04064615069188597893488373425, 2.21877903662779709649506577429, 2.26938920693951858837458204791, 2.31878033433103298259082609196, 2.44828283964513632151207691604, 2.60110875351930158529905306754, 2.60738705040407875798683767262, 2.79449494664242848019303923963, 2.86034621660371860889836957176, 3.03203500587748826523610814656, 3.03786233257994063421817908794
Plot not available for L-functions of degree greater than 10.
