| L(s) = 1 | + 4·2-s + 16·4-s + 26·5-s + 64·8-s + 104·10-s − 664·11-s − 318·13-s + 256·16-s + 1.58e3·17-s − 236·19-s + 416·20-s − 2.65e3·22-s − 2.21e3·23-s − 2.44e3·25-s − 1.27e3·26-s + 4.95e3·29-s + 7.12e3·31-s + 1.02e3·32-s + 6.32e3·34-s + 4.35e3·37-s − 944·38-s + 1.66e3·40-s + 1.05e4·41-s − 8.45e3·43-s − 1.06e4·44-s − 8.84e3·46-s + 5.35e3·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.465·5-s + 0.353·8-s + 0.328·10-s − 1.65·11-s − 0.521·13-s + 1/4·16-s + 1.32·17-s − 0.149·19-s + 0.232·20-s − 1.16·22-s − 0.871·23-s − 0.783·25-s − 0.369·26-s + 1.09·29-s + 1.33·31-s + 0.176·32-s + 0.938·34-s + 0.523·37-s − 0.106·38-s + 0.164·40-s + 0.979·41-s − 0.697·43-s − 0.827·44-s − 0.616·46-s + 0.353·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.562453157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.562453157\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 26 T + p^{5} T^{2} \) |
| 11 | \( 1 + 664 T + p^{5} T^{2} \) |
| 13 | \( 1 + 318 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1582 T + p^{5} T^{2} \) |
| 19 | \( 1 + 236 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2212 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4954 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7128 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4358 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10542 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8452 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5352 T + p^{5} T^{2} \) |
| 53 | \( 1 - 33354 T + p^{5} T^{2} \) |
| 59 | \( 1 + 15436 T + p^{5} T^{2} \) |
| 61 | \( 1 - 36762 T + p^{5} T^{2} \) |
| 67 | \( 1 - 40972 T + p^{5} T^{2} \) |
| 71 | \( 1 - 9092 T + p^{5} T^{2} \) |
| 73 | \( 1 - 73454 T + p^{5} T^{2} \) |
| 79 | \( 1 - 89400 T + p^{5} T^{2} \) |
| 83 | \( 1 + 6428 T + p^{5} T^{2} \) |
| 89 | \( 1 + 122658 T + p^{5} T^{2} \) |
| 97 | \( 1 + 21370 T + p^{5} T^{2} \) |
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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
−9.830204412266161579588226374633, −8.200904634741764565590055787780, −7.79519147296136066151975152302, −6.66413902296177698957089246806, −5.67832781002998429403441765829, −5.16296511657495689373863409054, −4.08586608337112018589326259123, −2.85577409699452078525149474351, −2.20943146850104180000624805886, −0.74042326125537727438094954709,
0.74042326125537727438094954709, 2.20943146850104180000624805886, 2.85577409699452078525149474351, 4.08586608337112018589326259123, 5.16296511657495689373863409054, 5.67832781002998429403441765829, 6.66413902296177698957089246806, 7.79519147296136066151975152302, 8.200904634741764565590055787780, 9.830204412266161579588226374633
