| L(s) = 1 | + 8.07e9·2-s + 2.82e15·3-s + 2.83e19·4-s − 4.30e22·5-s + 2.28e25·6-s − 4.31e27·7-s − 6.90e28·8-s − 2.30e30·9-s − 3.48e32·10-s − 3.09e33·11-s + 8.01e34·12-s − 1.30e36·13-s − 3.48e37·14-s − 1.21e38·15-s − 1.60e39·16-s + 1.01e40·17-s − 1.86e40·18-s + 6.77e41·19-s − 1.22e42·20-s − 1.22e43·21-s − 2.49e43·22-s + 2.44e44·23-s − 1.95e44·24-s − 8.52e44·25-s − 1.05e46·26-s − 3.56e46·27-s − 1.22e47·28-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.880·3-s + 0.768·4-s − 0.827·5-s + 1.17·6-s − 1.47·7-s − 0.308·8-s − 0.223·9-s − 1.10·10-s − 0.441·11-s + 0.676·12-s − 0.815·13-s − 1.96·14-s − 0.729·15-s − 1.17·16-s + 1.03·17-s − 0.297·18-s + 1.86·19-s − 0.635·20-s − 1.30·21-s − 0.587·22-s + 1.35·23-s − 0.271·24-s − 0.314·25-s − 1.08·26-s − 1.07·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(66-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+65/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(33)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{67}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 8.07e9T + 3.68e19T^{2} \) |
| 3 | \( 1 - 2.82e15T + 1.03e31T^{2} \) |
| 5 | \( 1 + 4.30e22T + 2.71e45T^{2} \) |
| 7 | \( 1 + 4.31e27T + 8.53e54T^{2} \) |
| 11 | \( 1 + 3.09e33T + 4.90e67T^{2} \) |
| 13 | \( 1 + 1.30e36T + 2.54e72T^{2} \) |
| 17 | \( 1 - 1.01e40T + 9.53e79T^{2} \) |
| 19 | \( 1 - 6.77e41T + 1.31e83T^{2} \) |
| 23 | \( 1 - 2.44e44T + 3.25e88T^{2} \) |
| 29 | \( 1 - 6.24e46T + 1.13e95T^{2} \) |
| 31 | \( 1 + 4.23e48T + 8.67e96T^{2} \) |
| 37 | \( 1 + 6.58e50T + 8.57e101T^{2} \) |
| 41 | \( 1 + 3.22e51T + 6.77e104T^{2} \) |
| 43 | \( 1 + 7.33e52T + 1.49e106T^{2} \) |
| 47 | \( 1 - 4.96e53T + 4.85e108T^{2} \) |
| 53 | \( 1 + 1.14e56T + 1.19e112T^{2} \) |
| 59 | \( 1 + 7.39e56T + 1.27e115T^{2} \) |
| 61 | \( 1 - 1.27e58T + 1.11e116T^{2} \) |
| 67 | \( 1 + 1.92e59T + 4.95e118T^{2} \) |
| 71 | \( 1 + 1.96e60T + 2.14e120T^{2} \) |
| 73 | \( 1 + 4.77e60T + 1.30e121T^{2} \) |
| 79 | \( 1 - 1.98e61T + 2.21e123T^{2} \) |
| 83 | \( 1 - 3.39e62T + 5.49e124T^{2} \) |
| 89 | \( 1 - 8.38e62T + 5.13e126T^{2} \) |
| 97 | \( 1 - 1.21e64T + 1.38e129T^{2} \) |
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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
−16.01013290913126232524332317795, −14.69919824163864471633351289661, −13.31708162118979439093500858992, −12.01639654150650562106414451255, −9.394016900306873436855339835438, −7.35151603075895830250849752515, −5.39121725880309566753437613863, −3.45666845193236614903521960548, −2.99313109284787755800306477961, 0,
2.99313109284787755800306477961, 3.45666845193236614903521960548, 5.39121725880309566753437613863, 7.35151603075895830250849752515, 9.394016900306873436855339835438, 12.01639654150650562106414451255, 13.31708162118979439093500858992, 14.69919824163864471633351289661, 16.01013290913126232524332317795
