L(s) = 1 | + 920·25-s − 560·37-s − 160·49-s − 5.44e3·109-s + 6.05e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.84e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | + 7.35·25-s − 2.48·37-s − 0.466·49-s − 4.78·109-s + 4.54·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 8.39·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(12.05524537\) |
\(L(\frac12)\) |
\(\approx\) |
\(12.05524537\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 80 T^{2} + 3294 p^{2} T^{4} + 80 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
good | 5 | \( ( 1 - 46 p T^{2} + 42378 T^{4} - 46 p^{7} T^{6} + p^{12} T^{8} )^{4} \) |
| 11 | \( ( 1 - 1514 T^{2} + 926634 T^{4} - 1514 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 13 | \( ( 1 - 4612 T^{2} + 11616054 T^{4} - 4612 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 17 | \( ( 1 - 5414 T^{2} + 55500234 T^{4} - 5414 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 19 | \( ( 1 + 848 p T^{2} + 153748398 T^{4} + 848 p^{7} T^{6} + p^{12} T^{8} )^{4} \) |
| 23 | \( ( 1 - 21338 T^{2} + 224925066 T^{4} - 21338 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 29 | \( ( 1 + 12020 T^{2} + 409245270 T^{4} + 12020 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 31 | \( ( 1 + 106460 T^{2} + 4606292934 T^{4} + 106460 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 37 | \( ( 1 + 70 T + 83658 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} )^{8} \) |
| 41 | \( ( 1 - 239462 T^{2} + 23511841290 T^{4} - 239462 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 43 | \( ( 1 - 279904 T^{2} + 32118357102 T^{4} - 279904 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 47 | \( ( 1 + 248252 T^{2} + 36269960262 T^{4} + 248252 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 53 | \( ( 1 - 101740 T^{2} + 16763162550 T^{4} - 101740 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 59 | \( ( 1 + 106124 T^{2} + 12887683638 T^{4} + 106124 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 61 | \( ( 1 - 141892 T^{2} + 14052119286 T^{4} - 141892 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 67 | \( ( 1 - 880288 T^{2} + 372059791374 T^{4} - 880288 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 71 | \( ( 1 - 1394330 T^{2} + 742236080010 T^{4} - 1394330 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 73 | \( ( 1 - 653092 T^{2} + 227462860326 T^{4} - 653092 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 79 | \( ( 1 - 1732144 T^{2} + 1235369637726 T^{4} - 1732144 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 83 | \( ( 1 + 240044 T^{2} - 314608170 T^{4} + 240044 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 89 | \( ( 1 - 2129366 T^{2} + 2125710459498 T^{4} - 2129366 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 97 | \( ( 1 - 1521220 T^{2} + 1874648017158 T^{4} - 1521220 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
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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
−2.07718290623219001704227874684, −1.99646247753033573095076040098, −1.95611125363345641816701402697, −1.84005704306634920919574886910, −1.78804841035393305291857095244, −1.75816474135724780392745278630, −1.69499995598373820860523244884, −1.68188398835101374159779744324, −1.49994538158732060011683412695, −1.32886842988780480347769980396, −1.30320615935478290923942930466, −1.25391539410915571278756029123, −1.02218527199848716887920868057, −0.977838021967609794065413069876, −0.950249878683502214674381304499, −0.905394288807118551260220096834, −0.870202430258331874061903953415, −0.842096346076015445912830944246, −0.67028114457894438504607793452, −0.63928422151465635228298453114, −0.37495536764191409760601457674, −0.35115296688372077481316765834, −0.18018034084218070779911730751, −0.16290340298251284226907612675, −0.11454076954381228600934608854,
0.11454076954381228600934608854, 0.16290340298251284226907612675, 0.18018034084218070779911730751, 0.35115296688372077481316765834, 0.37495536764191409760601457674, 0.63928422151465635228298453114, 0.67028114457894438504607793452, 0.842096346076015445912830944246, 0.870202430258331874061903953415, 0.905394288807118551260220096834, 0.950249878683502214674381304499, 0.977838021967609794065413069876, 1.02218527199848716887920868057, 1.25391539410915571278756029123, 1.30320615935478290923942930466, 1.32886842988780480347769980396, 1.49994538158732060011683412695, 1.68188398835101374159779744324, 1.69499995598373820860523244884, 1.75816474135724780392745278630, 1.78804841035393305291857095244, 1.84005704306634920919574886910, 1.95611125363345641816701402697, 1.99646247753033573095076040098, 2.07718290623219001704227874684
Plot not available for L-functions of degree greater than 10.