| L(s) = 1 | − 1.04e5·2-s + 2.08e7·3-s + 2.35e9·4-s − 2.17e12·6-s + 5.34e13·7-s + 6.52e14·8-s − 5.12e15·9-s − 1.47e17·11-s + 4.90e16·12-s + 4.54e18·13-s − 5.59e18·14-s − 8.84e19·16-s + 1.04e19·17-s + 5.36e20·18-s − 4.19e20·19-s + 1.11e21·21-s + 1.53e22·22-s + 6.05e21·23-s + 1.35e22·24-s − 4.75e23·26-s − 2.22e23·27-s + 1.25e23·28-s − 1.89e24·29-s + 5.96e24·31-s + 3.65e24·32-s − 3.06e24·33-s − 1.09e24·34-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.279·3-s + 0.274·4-s − 0.315·6-s + 0.608·7-s + 0.819·8-s − 0.922·9-s − 0.965·11-s + 0.0765·12-s + 1.89·13-s − 0.686·14-s − 1.19·16-s + 0.0521·17-s + 1.04·18-s − 0.333·19-s + 0.169·21-s + 1.08·22-s + 0.205·23-s + 0.228·24-s − 2.14·26-s − 0.536·27-s + 0.166·28-s − 1.40·29-s + 1.47·31-s + 0.534·32-s − 0.269·33-s − 0.0589·34-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 | \( 1 + 1.04e5T + 8.58e9T^{2} \) |
| 3 | \( 1 - 2.08e7T + 5.55e15T^{2} \) |
| 7 | \( 1 - 5.34e13T + 7.73e27T^{2} \) |
| 11 | \( 1 + 1.47e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 4.54e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 1.04e19T + 4.02e40T^{2} \) |
| 19 | \( 1 + 4.19e20T + 1.58e42T^{2} \) |
| 23 | \( 1 - 6.05e21T + 8.65e44T^{2} \) |
| 29 | \( 1 + 1.89e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 5.96e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 1.42e24T + 5.63e51T^{2} \) |
| 41 | \( 1 - 1.10e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.10e26T + 8.02e53T^{2} \) |
| 47 | \( 1 + 3.64e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 2.17e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 1.45e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 2.49e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 2.23e30T + 1.82e60T^{2} \) |
| 71 | \( 1 + 2.53e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 5.58e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 1.88e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 7.98e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.56e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 1.05e33T + 3.65e65T^{2} \) |
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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
−10.70331987173469502030898900651, −9.316963212265587865221209721149, −8.345634083849360148187074685119, −7.88128638467702132935835915566, −6.19765403007969689039416410828, −4.90074127765009345043262089941, −3.47575119649016919270232439770, −2.10758736601115251441302221519, −1.06705696230099061144674487677, 0,
1.06705696230099061144674487677, 2.10758736601115251441302221519, 3.47575119649016919270232439770, 4.90074127765009345043262089941, 6.19765403007969689039416410828, 7.88128638467702132935835915566, 8.345634083849360148187074685119, 9.316963212265587865221209721149, 10.70331987173469502030898900651
