| L(s) = 1 | + 4·2-s + 12·4-s − 7·5-s + 32·8-s − 28·10-s + 25·11-s − 49·13-s + 80·16-s − 98·17-s − 119·19-s − 84·20-s + 100·22-s − 122·23-s − 165·25-s − 196·26-s − 73·29-s + 98·31-s + 192·32-s − 392·34-s + 289·37-s − 476·38-s − 224·40-s − 336·41-s + 307·43-s + 300·44-s − 488·46-s − 672·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.626·5-s + 1.41·8-s − 0.885·10-s + 0.685·11-s − 1.04·13-s + 5/4·16-s − 1.39·17-s − 1.43·19-s − 0.939·20-s + 0.969·22-s − 1.10·23-s − 1.31·25-s − 1.47·26-s − 0.467·29-s + 0.567·31-s + 1.06·32-s − 1.97·34-s + 1.28·37-s − 2.03·38-s − 0.885·40-s − 1.27·41-s + 1.08·43-s + 1.02·44-s − 1.56·46-s − 2.08·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 7 T + 214 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 25 T + 454 T^{2} - 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 49 T + 3788 T^{2} + 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 98 T + 7402 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 119 T + 16824 T^{2} + 119 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 122 T + 18598 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 73 T + 47746 T^{2} + 73 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 98 T + 24155 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 289 T + 100908 T^{2} - 289 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 168 T + p^{3} T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 307 T + 161298 T^{2} - 307 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 672 T + 292750 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 375 T + 141406 T^{2} - 375 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 763 T + 376762 T^{2} + 763 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 406 T + 439394 T^{2} + 406 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1041 T + 813340 T^{2} + 1041 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1652 T + 1360270 T^{2} + 1652 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 189 T + 332980 T^{2} + 189 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 524 T + 591329 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 287 T + 781978 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2394 T + 2702050 T^{2} + 2394 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 63 T + 667132 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.426656846252583507829530561966, −9.330149609011566739577237593945, −8.460291241831101295802445413923, −8.282405845033937796974126537100, −7.56561073402363566116041252355, −7.48577880039717727204676446426, −6.64412150997610083310075131701, −6.59415154648269263109643904217, −5.90666844954223240648343957521, −5.76297600203475637077048581719, −4.86801640856186357835709615413, −4.44027529517803970643855491784, −4.16256002873271091903360136028, −3.97242035748965731568326301864, −2.96751064038971173380436658843, −2.73284251713205180685729574392, −1.84381105662550035174030615617, −1.64559667598509747989666703614, 0, 0,
1.64559667598509747989666703614, 1.84381105662550035174030615617, 2.73284251713205180685729574392, 2.96751064038971173380436658843, 3.97242035748965731568326301864, 4.16256002873271091903360136028, 4.44027529517803970643855491784, 4.86801640856186357835709615413, 5.76297600203475637077048581719, 5.90666844954223240648343957521, 6.59415154648269263109643904217, 6.64412150997610083310075131701, 7.48577880039717727204676446426, 7.56561073402363566116041252355, 8.282405845033937796974126537100, 8.460291241831101295802445413923, 9.330149609011566739577237593945, 9.426656846252583507829530561966
