| L(s) = 1 | + 8·2-s + 64·4-s + 135·5-s + 71·7-s + 512·8-s + 1.08e3·10-s + 543·11-s − 3.83e3·13-s + 568·14-s + 4.09e3·16-s − 3.56e4·17-s + 6.85e3·19-s + 8.64e3·20-s + 4.34e3·22-s + 1.03e4·23-s − 5.99e4·25-s − 3.07e4·26-s + 4.54e3·28-s − 2.48e5·29-s + 1.05e5·31-s + 3.27e4·32-s − 2.85e5·34-s + 9.58e3·35-s + 2.45e4·37-s + 5.48e4·38-s + 6.91e4·40-s + 7.12e5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.482·5-s + 0.0782·7-s + 0.353·8-s + 0.341·10-s + 0.123·11-s − 0.484·13-s + 0.0553·14-s + 1/4·16-s − 1.76·17-s + 0.229·19-s + 0.241·20-s + 0.0869·22-s + 0.176·23-s − 0.766·25-s − 0.342·26-s + 0.0391·28-s − 1.89·29-s + 0.634·31-s + 0.176·32-s − 1.24·34-s + 0.0377·35-s + 0.0796·37-s + 0.162·38-s + 0.170·40-s + 1.61·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 - 27 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 71 T + p^{7} T^{2} \) |
| 11 | \( 1 - 543 T + p^{7} T^{2} \) |
| 13 | \( 1 + 3838 T + p^{7} T^{2} \) |
| 17 | \( 1 + 35691 T + p^{7} T^{2} \) |
| 23 | \( 1 - 10308 T + p^{7} T^{2} \) |
| 29 | \( 1 + 248670 T + p^{7} T^{2} \) |
| 31 | \( 1 - 105242 T + p^{7} T^{2} \) |
| 37 | \( 1 - 24536 T + p^{7} T^{2} \) |
| 41 | \( 1 - 712008 T + p^{7} T^{2} \) |
| 43 | \( 1 + 539893 T + p^{7} T^{2} \) |
| 47 | \( 1 + 183231 T + p^{7} T^{2} \) |
| 53 | \( 1 + 589242 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2722680 T + p^{7} T^{2} \) |
| 61 | \( 1 + 3186073 T + p^{7} T^{2} \) |
| 67 | \( 1 - 702176 T + p^{7} T^{2} \) |
| 71 | \( 1 - 3058428 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4798483 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1638440 T + p^{7} T^{2} \) |
| 83 | \( 1 + 997332 T + p^{7} T^{2} \) |
| 89 | \( 1 + 8572890 T + p^{7} T^{2} \) |
| 97 | \( 1 + 185194 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.868900823498252085480584016082, −9.054498283400349122518321081170, −7.79797626759191364596637445732, −6.79444402100676782310486979299, −5.91256388415275175276307057418, −4.88695946935762644338429072391, −3.92800376040899476266150045374, −2.58492104220481390180595944881, −1.68317106919808438888480782907, 0,
1.68317106919808438888480782907, 2.58492104220481390180595944881, 3.92800376040899476266150045374, 4.88695946935762644338429072391, 5.91256388415275175276307057418, 6.79444402100676782310486979299, 7.79797626759191364596637445732, 9.054498283400349122518321081170, 9.868900823498252085480584016082
