| L(s) = 1 | + 1.62e5·2-s − 1.50e7·3-s + 1.76e10·4-s − 2.43e12·6-s + 1.33e14·7-s + 1.46e15·8-s − 5.33e15·9-s + 2.82e17·11-s − 2.65e17·12-s − 1.43e18·13-s + 2.16e19·14-s + 8.63e19·16-s + 2.38e20·17-s − 8.63e20·18-s − 5.86e20·19-s − 2.01e21·21-s + 4.56e22·22-s + 1.73e22·23-s − 2.21e22·24-s − 2.32e23·26-s + 1.64e23·27-s + 2.36e24·28-s − 1.13e23·29-s + 2.12e24·31-s + 1.36e24·32-s − 4.24e24·33-s + 3.86e25·34-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.201·3-s + 2.05·4-s − 0.353·6-s + 1.52·7-s + 1.84·8-s − 0.959·9-s + 1.85·11-s − 0.415·12-s − 0.597·13-s + 2.65·14-s + 1.16·16-s + 1.18·17-s − 1.67·18-s − 0.466·19-s − 0.307·21-s + 3.23·22-s + 0.589·23-s − 0.372·24-s − 1.04·26-s + 0.395·27-s + 3.12·28-s − 0.0841·29-s + 0.523·31-s + 0.199·32-s − 0.373·33-s + 2.07·34-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
| \(L(17)\) |
\(\approx\) |
\(9.065558843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.065558843\) |
| \(L(\frac{35}{2})\) |
|
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 | \( 1 - 1.62e5T + 8.58e9T^{2} \) |
| 3 | \( 1 + 1.50e7T + 5.55e15T^{2} \) |
| 7 | \( 1 - 1.33e14T + 7.73e27T^{2} \) |
| 11 | \( 1 - 2.82e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 1.43e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 2.38e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 5.86e20T + 1.58e42T^{2} \) |
| 23 | \( 1 - 1.73e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 1.13e23T + 1.81e48T^{2} \) |
| 31 | \( 1 - 2.12e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 8.11e25T + 5.63e51T^{2} \) |
| 41 | \( 1 + 1.96e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 8.68e26T + 8.02e53T^{2} \) |
| 47 | \( 1 - 5.03e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 7.43e27T + 7.96e56T^{2} \) |
| 59 | \( 1 - 5.72e28T + 2.74e58T^{2} \) |
| 61 | \( 1 + 2.92e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 8.52e29T + 1.82e60T^{2} \) |
| 71 | \( 1 + 1.52e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 1.66e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 2.86e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 1.27e30T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.21e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 9.03e32T + 3.65e65T^{2} \) |
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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
−11.81427391916756985770158457619, −10.81265689291909082036725674701, −8.831125584007940915568933215511, −7.39911581496801437059880098970, −6.18677702556556069409320919333, −5.25402941655093203394469541805, −4.41106147089835107709520383335, −3.37316426867054053034619568172, −2.12404540782365705275068795620, −1.08943558853637516336192538103,
1.08943558853637516336192538103, 2.12404540782365705275068795620, 3.37316426867054053034619568172, 4.41106147089835107709520383335, 5.25402941655093203394469541805, 6.18677702556556069409320919333, 7.39911581496801437059880098970, 8.831125584007940915568933215511, 10.81265689291909082036725674701, 11.81427391916756985770158457619
