| L(s) = 1 | − 1.09e10·2-s + 4.46e15·3-s + 8.20e19·4-s + 5.92e22·5-s − 4.86e25·6-s − 3.00e27·7-s − 4.92e29·8-s + 9.59e30·9-s − 6.46e32·10-s − 6.29e33·11-s + 3.66e35·12-s − 7.76e35·13-s + 3.28e37·14-s + 2.64e38·15-s + 2.34e39·16-s − 6.14e39·17-s − 1.04e41·18-s − 5.04e41·19-s + 4.86e42·20-s − 1.34e43·21-s + 6.86e43·22-s + 1.38e43·23-s − 2.19e45·24-s + 7.98e44·25-s + 8.46e45·26-s − 3.13e45·27-s − 2.46e47·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 1.38·3-s + 2.22·4-s + 1.13·5-s − 2.49·6-s − 1.02·7-s − 2.19·8-s + 0.931·9-s − 2.04·10-s − 0.898·11-s + 3.09·12-s − 0.486·13-s + 1.84·14-s + 1.58·15-s + 1.72·16-s − 0.629·17-s − 1.67·18-s − 1.39·19-s + 2.53·20-s − 1.43·21-s + 1.61·22-s + 0.0766·23-s − 3.05·24-s + 0.294·25-s + 0.873·26-s − 0.0948·27-s − 2.29·28-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 | \( 1 + 1.09e10T + 3.68e19T^{2} \) |
| 3 | \( 1 - 4.46e15T + 1.03e31T^{2} \) |
| 5 | \( 1 - 5.92e22T + 2.71e45T^{2} \) |
| 7 | \( 1 + 3.00e27T + 8.53e54T^{2} \) |
| 11 | \( 1 + 6.29e33T + 4.90e67T^{2} \) |
| 13 | \( 1 + 7.76e35T + 2.54e72T^{2} \) |
| 17 | \( 1 + 6.14e39T + 9.53e79T^{2} \) |
| 19 | \( 1 + 5.04e41T + 1.31e83T^{2} \) |
| 23 | \( 1 - 1.38e43T + 3.25e88T^{2} \) |
| 29 | \( 1 + 2.65e47T + 1.13e95T^{2} \) |
| 31 | \( 1 - 5.51e48T + 8.67e96T^{2} \) |
| 37 | \( 1 - 1.65e50T + 8.57e101T^{2} \) |
| 41 | \( 1 + 1.49e52T + 6.77e104T^{2} \) |
| 43 | \( 1 + 1.75e53T + 1.49e106T^{2} \) |
| 47 | \( 1 - 4.83e53T + 4.85e108T^{2} \) |
| 53 | \( 1 - 3.42e55T + 1.19e112T^{2} \) |
| 59 | \( 1 + 6.23e57T + 1.27e115T^{2} \) |
| 61 | \( 1 - 9.92e57T + 1.11e116T^{2} \) |
| 67 | \( 1 - 1.42e59T + 4.95e118T^{2} \) |
| 71 | \( 1 + 1.36e60T + 2.14e120T^{2} \) |
| 73 | \( 1 - 6.63e60T + 1.30e121T^{2} \) |
| 79 | \( 1 - 9.29e58T + 2.21e123T^{2} \) |
| 83 | \( 1 + 5.07e61T + 5.49e124T^{2} \) |
| 89 | \( 1 + 5.42e62T + 5.13e126T^{2} \) |
| 97 | \( 1 - 2.04e64T + 1.38e129T^{2} \) |
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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
−17.12694489467884750074997408029, −15.37804986430790404376850594061, −13.29435217272245598688037397761, −10.23928718747558885845957465912, −9.380069905523076979387573648353, −8.206391305087962312649636486345, −6.55543719167963494662796228037, −2.75043297436361265178081020797, −1.97359949007865296934919061665, 0,
1.97359949007865296934919061665, 2.75043297436361265178081020797, 6.55543719167963494662796228037, 8.206391305087962312649636486345, 9.380069905523076979387573648353, 10.23928718747558885845957465912, 13.29435217272245598688037397761, 15.37804986430790404376850594061, 17.12694489467884750074997408029
