| L(s) = 1 | − 2·2-s + 3-s − 5·5-s − 2·6-s + 8·8-s + 27·9-s + 10·10-s + 30·11-s − 88·13-s − 5·15-s − 16·16-s − 24·17-s − 54·18-s + 2·19-s − 60·22-s + 183·23-s + 8·24-s + 176·26-s + 80·27-s − 558·29-s + 10·30-s − 40·31-s + 30·33-s + 48·34-s + 76·37-s − 4·38-s − 88·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.192·3-s − 0.447·5-s − 0.136·6-s + 0.353·8-s + 9-s + 0.316·10-s + 0.822·11-s − 1.87·13-s − 0.0860·15-s − 1/4·16-s − 0.342·17-s − 0.707·18-s + 0.0241·19-s − 0.581·22-s + 1.65·23-s + 0.0680·24-s + 1.32·26-s + 0.570·27-s − 3.57·29-s + 0.0608·30-s − 0.231·31-s + 0.158·33-s + 0.242·34-s + 0.337·37-s − 0.0170·38-s − 0.361·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.747383938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.747383938\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 26 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 44 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 24 T - 4337 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 6855 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 183 T + 21322 T^{2} - 183 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 279 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 40 T - 28191 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 76 T - 44877 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 423 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 305 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 456 T + 104113 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 198 T - 109673 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 462 T + 8065 T^{2} + 462 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 281 T - 148020 T^{2} - 281 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 499 T - 51762 T^{2} - 499 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 534 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 800 T + 250983 T^{2} - 800 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10 p T + 21 p^{2} T^{2} - 10 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 597 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1017 T + 329320 T^{2} - 1017 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1330 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.75084626895147568401192001687, −10.32518438415436770850063077534, −9.527875560953472969215247232845, −9.461617389264161048085118159428, −9.023110082575065413377239806105, −8.935508423331263187411812707972, −7.77695007936298772975258649402, −7.50673423936608684727411219563, −7.35104814680081362124524197140, −7.08144549490628628892082913864, −6.08908900613563622444054560709, −5.76354640861375690862012314127, −4.78647706848393306437332377119, −4.68515249516828348051786487033, −3.87473829206685539547266960307, −3.60319258046312638790252756490, −2.31751795076146603807148791651, −2.26752007595557743914405475939, −1.05854981539769790488504046915, −0.55833354501674531324658755565,
0.55833354501674531324658755565, 1.05854981539769790488504046915, 2.26752007595557743914405475939, 2.31751795076146603807148791651, 3.60319258046312638790252756490, 3.87473829206685539547266960307, 4.68515249516828348051786487033, 4.78647706848393306437332377119, 5.76354640861375690862012314127, 6.08908900613563622444054560709, 7.08144549490628628892082913864, 7.35104814680081362124524197140, 7.50673423936608684727411219563, 7.77695007936298772975258649402, 8.935508423331263187411812707972, 9.023110082575065413377239806105, 9.461617389264161048085118159428, 9.527875560953472969215247232845, 10.32518438415436770850063077534, 10.75084626895147568401192001687
