| L(s) = 1 | − 4·2-s + 12·4-s − 12·5-s + 16·7-s − 32·8-s + 48·10-s − 36·11-s + 10·13-s − 64·14-s + 80·16-s − 120·17-s − 8·19-s − 144·20-s + 144·22-s − 180·23-s − 115·25-s − 40·26-s + 192·28-s − 324·29-s − 248·31-s − 192·32-s + 480·34-s − 192·35-s − 434·37-s + 32·38-s + 384·40-s − 192·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.07·5-s + 0.863·7-s − 1.41·8-s + 1.51·10-s − 0.986·11-s + 0.213·13-s − 1.22·14-s + 5/4·16-s − 1.71·17-s − 0.0965·19-s − 1.60·20-s + 1.39·22-s − 1.63·23-s − 0.919·25-s − 0.301·26-s + 1.29·28-s − 2.07·29-s − 1.43·31-s − 1.06·32-s + 2.42·34-s − 0.927·35-s − 1.92·37-s + 0.136·38-s + 1.51·40-s − 0.731·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 12 T + 259 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 642 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 36 T + 1258 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 T - 873 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 120 T + 10159 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 666 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 180 T + 32002 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 324 T + 73699 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 p T + 39966 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 434 T + 148287 T^{2} + 434 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 192 T + 94786 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 608 T + 250458 T^{2} + 608 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 384 T + 209518 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 408 T + 183418 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1008 T + 637126 T^{2} - 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 742 T + 9555 p T^{2} - 742 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 104 T + 90042 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1140 T + 1038994 T^{2} - 1140 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 850 T + 709827 T^{2} - 850 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 440 T + 966978 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 264 T + 1063798 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 768 T + 1343527 T^{2} + 768 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 764 T + 1886598 T^{2} + 764 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.67755049003422031342868460763, −11.55147484404414368394701485099, −11.18352807579592093243217712706, −10.69254113223151628980573646813, −10.03914823785766153218731762723, −9.731438335001040107315040395565, −8.771067104092293052342967882118, −8.568304504795535210966827679765, −8.043930522295126479344748887877, −7.67273223137470657373013128266, −7.05632802395289902446835256584, −6.62877909578204508362771540617, −5.52454323165698680593592555200, −5.21620566087277160546712159763, −3.89644642041884438646218258806, −3.71943993641938030771600832543, −2.04871865276272112224892157250, −1.98043388523693271977403590444, 0, 0,
1.98043388523693271977403590444, 2.04871865276272112224892157250, 3.71943993641938030771600832543, 3.89644642041884438646218258806, 5.21620566087277160546712159763, 5.52454323165698680593592555200, 6.62877909578204508362771540617, 7.05632802395289902446835256584, 7.67273223137470657373013128266, 8.043930522295126479344748887877, 8.568304504795535210966827679765, 8.771067104092293052342967882118, 9.731438335001040107315040395565, 10.03914823785766153218731762723, 10.69254113223151628980573646813, 11.18352807579592093243217712706, 11.55147484404414368394701485099, 11.67755049003422031342868460763
