| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 5·5-s − 6·6-s − 13·7-s − 8·8-s + 9·9-s − 10·10-s − 15·11-s + 12·12-s + 13·13-s + 26·14-s + 15·15-s + 16·16-s − 75·17-s − 18·18-s − 130·19-s + 20·20-s − 39·21-s + 30·22-s + 45·23-s − 24·24-s + 25·25-s − 26·26-s + 27·27-s − 52·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.701·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.411·11-s + 0.288·12-s + 0.277·13-s + 0.496·14-s + 0.258·15-s + 1/4·16-s − 1.07·17-s − 0.235·18-s − 1.56·19-s + 0.223·20-s − 0.405·21-s + 0.290·22-s + 0.407·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.350·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 7 | \( 1 + 13 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 17 | \( 1 + 75 T + p^{3} T^{2} \) |
| 19 | \( 1 + 130 T + p^{3} T^{2} \) |
| 23 | \( 1 - 45 T + p^{3} T^{2} \) |
| 29 | \( 1 + 138 T + p^{3} T^{2} \) |
| 31 | \( 1 + 34 T + p^{3} T^{2} \) |
| 37 | \( 1 + 379 T + p^{3} T^{2} \) |
| 41 | \( 1 - 243 T + p^{3} T^{2} \) |
| 43 | \( 1 - 416 T + p^{3} T^{2} \) |
| 47 | \( 1 - 378 T + p^{3} T^{2} \) |
| 53 | \( 1 + 3 T + p^{3} T^{2} \) |
| 59 | \( 1 + 816 T + p^{3} T^{2} \) |
| 61 | \( 1 + 607 T + p^{3} T^{2} \) |
| 67 | \( 1 + 700 T + p^{3} T^{2} \) |
| 71 | \( 1 - 57 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1162 T + p^{3} T^{2} \) |
| 79 | \( 1 + T + p^{3} T^{2} \) |
| 83 | \( 1 - 672 T + p^{3} T^{2} \) |
| 89 | \( 1 - 969 T + p^{3} T^{2} \) |
| 97 | \( 1 + 949 T + p^{3} T^{2} \) |
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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
−10.49746217776190218249311450062, −9.167627627909269886721340304860, −8.958240472993383321724222193928, −7.72691340841588252227533406787, −6.75718508440419394584933015533, −5.88095723867238428202706142257, −4.28226180805879219393266164906, −2.90267095188749439000892491153, −1.85149761686507112539677105631, 0,
1.85149761686507112539677105631, 2.90267095188749439000892491153, 4.28226180805879219393266164906, 5.88095723867238428202706142257, 6.75718508440419394584933015533, 7.72691340841588252227533406787, 8.958240472993383321724222193928, 9.167627627909269886721340304860, 10.49746217776190218249311450062
