| L(s) = 1 | − 7.64e4·2-s − 7.43e7·3-s − 2.75e9·4-s + 5.67e12·6-s − 1.36e14·7-s + 8.66e14·8-s − 3.57e13·9-s + 3.84e16·11-s + 2.04e17·12-s − 2.57e18·13-s + 1.04e19·14-s − 4.25e19·16-s − 1.44e20·17-s + 2.73e18·18-s − 2.79e20·19-s + 1.01e22·21-s − 2.93e21·22-s − 3.71e22·23-s − 6.44e22·24-s + 1.96e23·26-s + 4.15e23·27-s + 3.76e23·28-s − 1.90e24·29-s + 5.88e24·31-s − 4.19e24·32-s − 2.85e24·33-s + 1.10e25·34-s + ⋯ |
| L(s) = 1 | − 0.824·2-s − 0.996·3-s − 0.320·4-s + 0.821·6-s − 1.55·7-s + 1.08·8-s − 0.00642·9-s + 0.252·11-s + 0.319·12-s − 1.07·13-s + 1.28·14-s − 0.577·16-s − 0.720·17-s + 0.00530·18-s − 0.222·19-s + 1.55·21-s − 0.207·22-s − 1.26·23-s − 1.08·24-s + 0.883·26-s + 1.00·27-s + 0.499·28-s − 1.41·29-s + 1.45·31-s − 0.612·32-s − 0.251·33-s + 0.594·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 7.64e4T + 8.58e9T^{2} \) |
| 3 | \( 1 + 7.43e7T + 5.55e15T^{2} \) |
| 7 | \( 1 + 1.36e14T + 7.73e27T^{2} \) |
| 11 | \( 1 - 3.84e16T + 2.32e34T^{2} \) |
| 13 | \( 1 + 2.57e18T + 5.75e36T^{2} \) |
| 17 | \( 1 + 1.44e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 2.79e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 3.71e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 1.90e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 5.88e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 1.05e26T + 5.63e51T^{2} \) |
| 41 | \( 1 + 2.64e25T + 1.66e53T^{2} \) |
| 43 | \( 1 + 1.20e27T + 8.02e53T^{2} \) |
| 47 | \( 1 - 3.53e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 3.25e28T + 7.96e56T^{2} \) |
| 59 | \( 1 - 3.41e28T + 2.74e58T^{2} \) |
| 61 | \( 1 + 6.01e28T + 8.23e58T^{2} \) |
| 67 | \( 1 - 2.17e30T + 1.82e60T^{2} \) |
| 71 | \( 1 - 9.20e29T + 1.23e61T^{2} \) |
| 73 | \( 1 - 4.72e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 2.14e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 6.60e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 6.08e31T + 2.13e64T^{2} \) |
| 97 | \( 1 + 6.12e30T + 3.65e65T^{2} \) |
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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
−10.37362819963241476821337249915, −9.726384623501412001220355446218, −8.630835084099307249216292041923, −7.13747322205519082188443558248, −6.19747540538908860307250248825, −4.99771487993506434747278710055, −3.73323099622937728265271118449, −2.19101088163560279725123140869, −0.57848570201606078914713113170, 0,
0.57848570201606078914713113170, 2.19101088163560279725123140869, 3.73323099622937728265271118449, 4.99771487993506434747278710055, 6.19747540538908860307250248825, 7.13747322205519082188443558248, 8.630835084099307249216292041923, 9.726384623501412001220355446218, 10.37362819963241476821337249915
