L(s) = 1 | + 3.48·2-s − 0.850·3-s + 4.14·4-s − 5·5-s − 2.96·6-s − 13.4·8-s − 26.2·9-s − 17.4·10-s − 6.90·11-s − 3.52·12-s + 22.1·13-s + 4.25·15-s − 79.9·16-s − 88.3·17-s − 91.5·18-s − 36.9·19-s − 20.7·20-s − 24.0·22-s − 95.5·23-s + 11.4·24-s + 25·25-s + 77.1·26-s + 45.2·27-s + 269.·29-s + 14.8·30-s − 197.·31-s − 171.·32-s + ⋯ |
L(s) = 1 | + 1.23·2-s − 0.163·3-s + 0.518·4-s − 0.447·5-s − 0.201·6-s − 0.593·8-s − 0.973·9-s − 0.551·10-s − 0.189·11-s − 0.0848·12-s + 0.472·13-s + 0.0731·15-s − 1.24·16-s − 1.25·17-s − 1.19·18-s − 0.446·19-s − 0.231·20-s − 0.233·22-s − 0.866·23-s + 0.0970·24-s + 0.200·25-s + 0.582·26-s + 0.322·27-s + 1.72·29-s + 0.0901·30-s − 1.14·31-s − 0.946·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 3.48T + 8T^{2} \) |
| 3 | \( 1 + 0.850T + 27T^{2} \) |
| 11 | \( 1 + 6.90T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 88.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 36.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 95.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 2.14T + 5.06e4T^{2} \) |
| 41 | \( 1 + 174.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 17.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 528.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 641.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 642.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 142.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 478.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 105.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 986.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 711.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 636.T + 9.12e5T^{2} \) |
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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
−11.46727357399783508430770319764, −10.60995594645421670388143769536, −9.032197471101293627578520611794, −8.278910882999519101777007002571, −6.72071123553706145178269695424, −5.86873427559710920902523044208, −4.77916952741064063075795841995, −3.78336931782859209529655010140, −2.55146116296370456192564060613, 0,
2.55146116296370456192564060613, 3.78336931782859209529655010140, 4.77916952741064063075795841995, 5.86873427559710920902523044208, 6.72071123553706145178269695424, 8.278910882999519101777007002571, 9.032197471101293627578520611794, 10.60995594645421670388143769536, 11.46727357399783508430770319764