| L(s) = 1 | − 6.12e4·2-s + 1.16e8·3-s − 4.83e9·4-s − 7.12e12·6-s + 1.52e14·7-s + 8.22e14·8-s + 7.98e15·9-s − 2.70e17·11-s − 5.63e17·12-s − 3.01e18·13-s − 9.35e18·14-s − 8.79e18·16-s − 5.13e19·17-s − 4.89e20·18-s − 2.04e21·19-s + 1.77e22·21-s + 1.65e22·22-s + 2.04e22·23-s + 9.57e22·24-s + 1.84e23·26-s + 2.82e23·27-s − 7.38e23·28-s + 1.42e24·29-s + 5.31e24·31-s − 6.52e24·32-s − 3.15e25·33-s + 3.14e24·34-s + ⋯ |
| L(s) = 1 | − 0.660·2-s + 1.56·3-s − 0.563·4-s − 1.03·6-s + 1.73·7-s + 1.03·8-s + 1.43·9-s − 1.77·11-s − 0.879·12-s − 1.25·13-s − 1.14·14-s − 0.119·16-s − 0.255·17-s − 0.949·18-s − 1.62·19-s + 2.71·21-s + 1.17·22-s + 0.694·23-s + 1.61·24-s + 0.830·26-s + 0.682·27-s − 0.978·28-s + 1.05·29-s + 1.31·31-s − 0.954·32-s − 2.77·33-s + 0.169·34-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
| \(L(17)\) |
\(\approx\) |
\(2.502902248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.502902248\) |
| \(L(\frac{35}{2})\) |
|
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 | \( 1 + 6.12e4T + 8.58e9T^{2} \) |
| 3 | \( 1 - 1.16e8T + 5.55e15T^{2} \) |
| 7 | \( 1 - 1.52e14T + 7.73e27T^{2} \) |
| 11 | \( 1 + 2.70e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 3.01e18T + 5.75e36T^{2} \) |
| 17 | \( 1 + 5.13e19T + 4.02e40T^{2} \) |
| 19 | \( 1 + 2.04e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 2.04e22T + 8.65e44T^{2} \) |
| 29 | \( 1 - 1.42e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 5.31e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 5.96e25T + 5.63e51T^{2} \) |
| 41 | \( 1 - 2.10e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 3.53e26T + 8.02e53T^{2} \) |
| 47 | \( 1 + 1.08e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 1.51e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 3.91e28T + 2.74e58T^{2} \) |
| 61 | \( 1 - 1.55e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 5.34e29T + 1.82e60T^{2} \) |
| 71 | \( 1 - 4.86e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 5.33e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 2.81e30T + 4.18e62T^{2} \) |
| 83 | \( 1 + 2.25e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 5.60e31T + 2.13e64T^{2} \) |
| 97 | \( 1 + 2.05e32T + 3.65e65T^{2} \) |
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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
−10.82898490911629434924648445328, −9.922813609602704928913024957137, −8.612839794249102915423951564492, −8.140702449568564028682439529955, −7.47079242349429566725085518591, −4.94390015419827740319514365604, −4.40966699450831363928393858884, −2.59858336442621186746402452125, −2.05603148450512807669540001784, −0.69551460691315709723118978861,
0.69551460691315709723118978861, 2.05603148450512807669540001784, 2.59858336442621186746402452125, 4.40966699450831363928393858884, 4.94390015419827740319514365604, 7.47079242349429566725085518591, 8.140702449568564028682439529955, 8.612839794249102915423951564492, 9.922813609602704928913024957137, 10.82898490911629434924648445328
