| L(s) = 1 | + 3.46·2-s + 3·3-s + 3.99·4-s − 5·5-s + 10.3·6-s + 27.7·7-s − 13.8·8-s + 9·9-s − 17.3·10-s + 11.9·12-s − 58.8·13-s + 95.9·14-s − 15·15-s − 80·16-s + 19.0·17-s + 31.1·18-s − 45.0·19-s − 19.9·20-s + 83.1·21-s − 75·23-s − 41.5·24-s + 25·25-s − 203.·26-s + 27·27-s + 110.·28-s − 128.·29-s − 51.9·30-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.499·4-s − 0.447·5-s + 0.707·6-s + 1.49·7-s − 0.612·8-s + 0.333·9-s − 0.547·10-s + 0.288·12-s − 1.25·13-s + 1.83·14-s − 0.258·15-s − 1.25·16-s + 0.271·17-s + 0.408·18-s − 0.543·19-s − 0.223·20-s + 0.863·21-s − 0.679·23-s − 0.353·24-s + 0.200·25-s − 1.53·26-s + 0.192·27-s + 0.748·28-s − 0.820·29-s − 0.316·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3.46T + 8T^{2} \) |
| 7 | \( 1 - 27.7T + 343T^{2} \) |
| 13 | \( 1 + 58.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 75T + 1.21e4T^{2} \) |
| 29 | \( 1 + 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 263T + 2.97e4T^{2} \) |
| 37 | \( 1 + 308T + 5.06e4T^{2} \) |
| 41 | \( 1 - 162.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 38.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 93T + 1.03e5T^{2} \) |
| 53 | \( 1 - 525T + 1.48e5T^{2} \) |
| 59 | \( 1 - 498T + 2.05e5T^{2} \) |
| 61 | \( 1 + 441.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 316T + 3.00e5T^{2} \) |
| 71 | \( 1 + 288T + 3.57e5T^{2} \) |
| 73 | \( 1 + 928.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 571.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 180T + 7.04e5T^{2} \) |
| 97 | \( 1 + 904T + 9.12e5T^{2} \) |
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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
−8.481055845634871242017171902337, −7.57737050200561367640296464145, −7.07385951830119899125341657325, −5.69466398987297751106240968308, −5.10526581403036087590839088111, −4.31552042648913352660920212143, −3.73139154377647754815279286811, −2.56593384136032162932678342029, −1.75485770745531347933769299760, 0,
1.75485770745531347933769299760, 2.56593384136032162932678342029, 3.73139154377647754815279286811, 4.31552042648913352660920212143, 5.10526581403036087590839088111, 5.69466398987297751106240968308, 7.07385951830119899125341657325, 7.57737050200561367640296464145, 8.481055845634871242017171902337
