| L(s) = 1 | + 4.29e9·2-s + 5.69e15·3-s + 1.84e19·4-s + 8.97e21·5-s + 2.44e25·6-s + 5.30e27·7-s + 7.92e28·8-s + 2.20e31·9-s + 3.85e31·10-s − 2.32e33·11-s + 1.04e35·12-s − 1.46e36·13-s + 2.27e37·14-s + 5.10e37·15-s + 3.40e38·16-s + 2.91e39·17-s + 9.49e40·18-s + 2.74e41·19-s + 1.65e41·20-s + 3.02e43·21-s − 9.97e42·22-s − 3.50e44·23-s + 4.50e44·24-s − 2.62e45·25-s − 6.31e45·26-s + 6.71e46·27-s + 9.78e46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.77·3-s + 0.5·4-s + 0.172·5-s + 1.25·6-s + 1.81·7-s + 0.353·8-s + 2.14·9-s + 0.121·10-s − 0.331·11-s + 0.886·12-s − 0.920·13-s + 1.28·14-s + 0.305·15-s + 0.250·16-s + 0.298·17-s + 1.51·18-s + 0.755·19-s + 0.0862·20-s + 3.22·21-s − 0.234·22-s − 1.94·23-s + 0.627·24-s − 0.970·25-s − 0.650·26-s + 2.03·27-s + 0.907·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+65/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(33)\) |
\(\approx\) |
\(7.671510385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.671510385\) |
| \(L(\frac{67}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.29e9T \) |
| good | 3 | \( 1 - 5.69e15T + 1.03e31T^{2} \) |
| 5 | \( 1 - 8.97e21T + 2.71e45T^{2} \) |
| 7 | \( 1 - 5.30e27T + 8.53e54T^{2} \) |
| 11 | \( 1 + 2.32e33T + 4.90e67T^{2} \) |
| 13 | \( 1 + 1.46e36T + 2.54e72T^{2} \) |
| 17 | \( 1 - 2.91e39T + 9.53e79T^{2} \) |
| 19 | \( 1 - 2.74e41T + 1.31e83T^{2} \) |
| 23 | \( 1 + 3.50e44T + 3.25e88T^{2} \) |
| 29 | \( 1 - 3.29e47T + 1.13e95T^{2} \) |
| 31 | \( 1 + 6.66e47T + 8.67e96T^{2} \) |
| 37 | \( 1 + 1.78e51T + 8.57e101T^{2} \) |
| 41 | \( 1 - 1.18e52T + 6.77e104T^{2} \) |
| 43 | \( 1 + 2.18e52T + 1.49e106T^{2} \) |
| 47 | \( 1 + 1.89e54T + 4.85e108T^{2} \) |
| 53 | \( 1 - 8.64e55T + 1.19e112T^{2} \) |
| 59 | \( 1 - 8.99e56T + 1.27e115T^{2} \) |
| 61 | \( 1 + 4.97e56T + 1.11e116T^{2} \) |
| 67 | \( 1 - 1.70e59T + 4.95e118T^{2} \) |
| 71 | \( 1 + 8.12e58T + 2.14e120T^{2} \) |
| 73 | \( 1 - 2.82e60T + 1.30e121T^{2} \) |
| 79 | \( 1 - 4.27e61T + 2.21e123T^{2} \) |
| 83 | \( 1 + 2.84e62T + 5.49e124T^{2} \) |
| 89 | \( 1 - 1.66e63T + 5.13e126T^{2} \) |
| 97 | \( 1 + 3.26e64T + 1.38e129T^{2} \) |
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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
−14.34722990239503002712507965752, −13.88244082295103877329927649407, −12.02302401125657699551334807788, −10.04212975486410247557076219410, −8.285150551462190852755256800285, −7.51824270436280006839472335201, −5.08626935017337662750911379832, −3.86007448495643775061002508817, −2.38542870140314331208010161254, −1.64727244612375915654148940377,
1.64727244612375915654148940377, 2.38542870140314331208010161254, 3.86007448495643775061002508817, 5.08626935017337662750911379832, 7.51824270436280006839472335201, 8.285150551462190852755256800285, 10.04212975486410247557076219410, 12.02302401125657699551334807788, 13.88244082295103877329927649407, 14.34722990239503002712507965752
