| 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
| \(L(17)\) |
\(\approx\) |
\(0.3185371362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3185371362\) |
| \(L(\frac{35}{2})\) |
|
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
−11.71920421232558148858469385912, −10.25678121633683334005910016236, −9.073586271580610120251079339826, −7.57700014013271477344843155732, −6.20446212864076688760849650352, −5.32774708354706613334495747059, −4.47024246608897488291176707449, −3.04896234977670088570140354089, −2.42190824930577917721275705662, −0.18970685829493163860544235682,
0.18970685829493163860544235682, 2.42190824930577917721275705662, 3.04896234977670088570140354089, 4.47024246608897488291176707449, 5.32774708354706613334495747059, 6.20446212864076688760849650352, 7.57700014013271477344843155732, 9.073586271580610120251079339826, 10.25678121633683334005910016236, 11.71920421232558148858469385912
