| L(s) = 1 | − 8.81e9·2-s + 1.85e15·3-s + 4.08e19·4-s − 7.15e22·5-s − 1.63e25·6-s + 3.06e27·7-s − 3.48e28·8-s + 3.43e30·9-s + 6.30e32·10-s − 1.25e34·11-s + 7.56e34·12-s + 2.31e36·13-s − 2.69e37·14-s − 1.32e38·15-s − 1.19e39·16-s − 7.49e39·17-s − 3.02e40·18-s + 5.84e41·19-s − 2.92e42·20-s + 5.67e42·21-s + 1.11e44·22-s + 9.17e43·23-s − 6.46e43·24-s + 2.40e45·25-s − 2.04e46·26-s + 6.36e45·27-s + 1.25e47·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 0.577·3-s + 1.10·4-s − 1.37·5-s − 0.838·6-s + 1.04·7-s − 0.155·8-s + 0.333·9-s + 1.99·10-s − 1.79·11-s + 0.639·12-s + 1.45·13-s − 1.52·14-s − 0.793·15-s − 0.881·16-s − 0.767·17-s − 0.483·18-s + 1.61·19-s − 1.52·20-s + 0.604·21-s + 2.61·22-s + 0.508·23-s − 0.0899·24-s + 0.887·25-s − 2.10·26-s + 0.192·27-s + 1.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+65/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(33)\) |
\(\approx\) |
\(0.8107746622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8107746622\) |
| \(L(\frac{67}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.85e15T \) |
| good | 2 | \( 1 + 8.81e9T + 3.68e19T^{2} \) |
| 5 | \( 1 + 7.15e22T + 2.71e45T^{2} \) |
| 7 | \( 1 - 3.06e27T + 8.53e54T^{2} \) |
| 11 | \( 1 + 1.25e34T + 4.90e67T^{2} \) |
| 13 | \( 1 - 2.31e36T + 2.54e72T^{2} \) |
| 17 | \( 1 + 7.49e39T + 9.53e79T^{2} \) |
| 19 | \( 1 - 5.84e41T + 1.31e83T^{2} \) |
| 23 | \( 1 - 9.17e43T + 3.25e88T^{2} \) |
| 29 | \( 1 + 6.88e46T + 1.13e95T^{2} \) |
| 31 | \( 1 + 2.32e48T + 8.67e96T^{2} \) |
| 37 | \( 1 + 8.82e50T + 8.57e101T^{2} \) |
| 41 | \( 1 - 4.03e51T + 6.77e104T^{2} \) |
| 43 | \( 1 + 1.44e53T + 1.49e106T^{2} \) |
| 47 | \( 1 + 2.32e53T + 4.85e108T^{2} \) |
| 53 | \( 1 - 7.19e55T + 1.19e112T^{2} \) |
| 59 | \( 1 + 6.15e57T + 1.27e115T^{2} \) |
| 61 | \( 1 + 2.93e56T + 1.11e116T^{2} \) |
| 67 | \( 1 + 1.10e59T + 4.95e118T^{2} \) |
| 71 | \( 1 - 9.28e59T + 2.14e120T^{2} \) |
| 73 | \( 1 + 8.71e59T + 1.30e121T^{2} \) |
| 79 | \( 1 + 3.00e61T + 2.21e123T^{2} \) |
| 83 | \( 1 - 3.36e62T + 5.49e124T^{2} \) |
| 89 | \( 1 - 3.18e63T + 5.13e126T^{2} \) |
| 97 | \( 1 - 5.11e64T + 1.38e129T^{2} \) |
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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
−13.47735162603644812208571685867, −11.42761276724687967289261978078, −10.61940106865304663929622479533, −8.839535614269563187871105898190, −8.002492020174373945899939633778, −7.41303313193377399493859810504, −4.87336752232567447310001262839, −3.32062737752976500789271382664, −1.75778697562675250265946578206, −0.56352291936085473274108862816,
0.56352291936085473274108862816, 1.75778697562675250265946578206, 3.32062737752976500789271382664, 4.87336752232567447310001262839, 7.41303313193377399493859810504, 8.002492020174373945899939633778, 8.839535614269563187871105898190, 10.61940106865304663929622479533, 11.42761276724687967289261978078, 13.47735162603644812208571685867
