L(s) = 1 | − 4.66·2-s + 13.7·4-s − 20.4·7-s − 26.6·8-s + 11·11-s + 75.4·13-s + 95.1·14-s + 14.4·16-s + 57.7·17-s − 27.5·19-s − 51.2·22-s + 125.·23-s − 351.·26-s − 279.·28-s − 73.6·29-s − 316.·31-s + 145.·32-s − 268.·34-s − 71.5·37-s + 128.·38-s − 359.·41-s − 380.·43-s + 150.·44-s − 585.·46-s + 576.·47-s + 73.4·49-s + 1.03e3·52-s + ⋯ |
L(s) = 1 | − 1.64·2-s + 1.71·4-s − 1.10·7-s − 1.17·8-s + 0.301·11-s + 1.61·13-s + 1.81·14-s + 0.225·16-s + 0.823·17-s − 0.332·19-s − 0.496·22-s + 1.13·23-s − 2.65·26-s − 1.88·28-s − 0.471·29-s − 1.83·31-s + 0.805·32-s − 1.35·34-s − 0.318·37-s + 0.547·38-s − 1.36·41-s − 1.34·43-s + 0.517·44-s − 1.87·46-s + 1.78·47-s + 0.214·49-s + 2.76·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
good | 2 | \( 1 + 4.66T + 8T^{2} \) |
| 7 | \( 1 + 20.4T + 343T^{2} \) |
| 13 | \( 1 - 75.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 73.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 316.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 71.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 359.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 380.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 576.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 252.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 552.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 9.30T + 2.26e5T^{2} \) |
| 67 | \( 1 + 610.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 132.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 220.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 9.99e2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 229.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 178.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 104.T + 9.12e5T^{2} \) |
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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
−8.482883363434380004609655528373, −7.53903036442406833773555651562, −6.84089423044659847099321138504, −6.26258945861292929571469287933, −5.33986671588517192722339809630, −3.77908289715107635310028291362, −3.17542190912253620478672965185, −1.84972059580280527118611495164, −1.01162658253182613210369896565, 0,
1.01162658253182613210369896565, 1.84972059580280527118611495164, 3.17542190912253620478672965185, 3.77908289715107635310028291362, 5.33986671588517192722339809630, 6.26258945861292929571469287933, 6.84089423044659847099321138504, 7.53903036442406833773555651562, 8.482883363434380004609655528373