| L(s) = 1 | − 1.78e5·2-s − 8.55e7·3-s + 2.34e10·4-s + 1.52e13·6-s + 8.09e13·7-s − 2.64e15·8-s + 1.75e15·9-s + 1.95e17·11-s − 2.00e18·12-s + 2.59e18·13-s − 1.44e19·14-s + 2.72e20·16-s + 1.52e20·17-s − 3.13e20·18-s + 1.39e21·19-s − 6.92e21·21-s − 3.50e22·22-s + 3.40e22·23-s + 2.26e23·24-s − 4.63e23·26-s + 3.25e23·27-s + 1.89e24·28-s − 1.49e24·29-s − 1.38e24·31-s − 2.60e25·32-s − 1.67e25·33-s − 2.72e25·34-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 1.14·3-s + 2.72·4-s + 2.21·6-s + 0.920·7-s − 3.32·8-s + 0.315·9-s + 1.28·11-s − 3.12·12-s + 1.08·13-s − 1.77·14-s + 3.69·16-s + 0.758·17-s − 0.608·18-s + 1.10·19-s − 1.05·21-s − 2.47·22-s + 1.15·23-s + 3.81·24-s − 2.08·26-s + 0.785·27-s + 2.50·28-s − 1.10·29-s − 0.341·31-s − 3.80·32-s − 1.47·33-s − 1.46·34-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 | \( 1 + 1.78e5T + 8.58e9T^{2} \) |
| 3 | \( 1 + 8.55e7T + 5.55e15T^{2} \) |
| 7 | \( 1 - 8.09e13T + 7.73e27T^{2} \) |
| 11 | \( 1 - 1.95e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 2.59e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 1.52e20T + 4.02e40T^{2} \) |
| 19 | \( 1 - 1.39e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 3.40e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 1.49e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 1.38e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 1.15e26T + 5.63e51T^{2} \) |
| 41 | \( 1 + 6.35e25T + 1.66e53T^{2} \) |
| 43 | \( 1 - 2.64e26T + 8.02e53T^{2} \) |
| 47 | \( 1 + 1.03e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 5.20e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 6.17e28T + 2.74e58T^{2} \) |
| 61 | \( 1 + 1.21e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 2.87e29T + 1.82e60T^{2} \) |
| 71 | \( 1 + 2.84e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 2.67e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 7.84e30T + 4.18e62T^{2} \) |
| 83 | \( 1 + 1.28e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 1.56e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 4.84e32T + 3.65e65T^{2} \) |
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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
−10.86108810698877534280840627566, −9.486992190160985777138808307952, −8.562146671451439324666811252519, −7.38381437118278793038202235615, −6.39032527180151610881806664758, −5.37251488569138577519494693180, −3.31551547423506106715895853466, −1.41828311182071710072888957568, −1.23898435765368584649546370056, 0,
1.23898435765368584649546370056, 1.41828311182071710072888957568, 3.31551547423506106715895853466, 5.37251488569138577519494693180, 6.39032527180151610881806664758, 7.38381437118278793038202235615, 8.562146671451439324666811252519, 9.486992190160985777138808307952, 10.86108810698877534280840627566
