| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 26·5-s + 36·6-s − 64·8-s + 81·9-s + 104·10-s − 358·11-s − 144·12-s − 332·13-s + 234·15-s + 256·16-s − 126·17-s − 324·18-s + 2.20e3·19-s − 416·20-s + 1.43e3·22-s − 2.14e3·23-s + 576·24-s − 2.44e3·25-s + 1.32e3·26-s − 729·27-s − 3.61e3·29-s − 936·30-s − 5.66e3·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.465·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.328·10-s − 0.892·11-s − 0.288·12-s − 0.544·13-s + 0.268·15-s + 1/4·16-s − 0.105·17-s − 0.235·18-s + 1.39·19-s − 0.232·20-s + 0.630·22-s − 0.844·23-s + 0.204·24-s − 0.783·25-s + 0.385·26-s − 0.192·27-s − 0.797·29-s − 0.189·30-s − 1.05·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6009647163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6009647163\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 26 T + p^{5} T^{2} \) |
| 11 | \( 1 + 358 T + p^{5} T^{2} \) |
| 13 | \( 1 + 332 T + p^{5} T^{2} \) |
| 17 | \( 1 + 126 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2200 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2142 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3610 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5668 T + p^{5} T^{2} \) |
| 37 | \( 1 + 2922 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2142 T + p^{5} T^{2} \) |
| 43 | \( 1 - 6388 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6520 T + p^{5} T^{2} \) |
| 53 | \( 1 + 10702 T + p^{5} T^{2} \) |
| 59 | \( 1 + 42524 T + p^{5} T^{2} \) |
| 61 | \( 1 - 44840 T + p^{5} T^{2} \) |
| 67 | \( 1 + 1448 T + p^{5} T^{2} \) |
| 71 | \( 1 + 62 p T + p^{5} T^{2} \) |
| 73 | \( 1 + 20500 T + p^{5} T^{2} \) |
| 79 | \( 1 - 65236 T + p^{5} T^{2} \) |
| 83 | \( 1 - 102804 T + p^{5} T^{2} \) |
| 89 | \( 1 - 128006 T + p^{5} T^{2} \) |
| 97 | \( 1 - 113324 T + p^{5} T^{2} \) |
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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
−10.89998889112775390931188380798, −10.01551586200578884775026638630, −9.177276495302374498963343358431, −7.77714911097757860906220121471, −7.41255817535828905180997149700, −5.99952106366225872595590896861, −5.03939370182261565538025126896, −3.55332159566161255377636934530, −2.05298204378893589144892360531, −0.48421193587887993428961725216,
0.48421193587887993428961725216, 2.05298204378893589144892360531, 3.55332159566161255377636934530, 5.03939370182261565538025126896, 5.99952106366225872595590896861, 7.41255817535828905180997149700, 7.77714911097757860906220121471, 9.177276495302374498963343358431, 10.01551586200578884775026638630, 10.89998889112775390931188380798
