| L(s) = 1 | − 2.43·2-s + 3.92·4-s − 5-s + 4.14·7-s − 4.67·8-s + 2.43·10-s + 3.68·11-s − 1.05·13-s − 10.0·14-s + 3.53·16-s + 1.65·17-s − 1.10·19-s − 3.92·20-s − 8.97·22-s + 1.38·23-s + 25-s + 2.57·26-s + 16.2·28-s + 4.45·29-s − 4.52·31-s + 0.742·32-s − 4.03·34-s − 4.14·35-s − 8.38·37-s + 2.68·38-s + 4.67·40-s − 3.02·41-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.96·4-s − 0.447·5-s + 1.56·7-s − 1.65·8-s + 0.769·10-s + 1.11·11-s − 0.293·13-s − 2.69·14-s + 0.884·16-s + 0.401·17-s − 0.252·19-s − 0.877·20-s − 1.91·22-s + 0.289·23-s + 0.200·25-s + 0.504·26-s + 3.07·28-s + 0.827·29-s − 0.812·31-s + 0.131·32-s − 0.691·34-s − 0.701·35-s − 1.37·37-s + 0.434·38-s + 0.739·40-s − 0.472·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.040101217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040101217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 13 | \( 1 + 1.05T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 19 | \( 1 + 1.10T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 - 4.45T + 29T^{2} \) |
| 31 | \( 1 + 4.52T + 31T^{2} \) |
| 37 | \( 1 + 8.38T + 37T^{2} \) |
| 41 | \( 1 + 3.02T + 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 + 3.44T + 53T^{2} \) |
| 59 | \( 1 - 0.568T + 59T^{2} \) |
| 61 | \( 1 - 1.53T + 61T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 - 0.771T + 71T^{2} \) |
| 73 | \( 1 - 9.97T + 73T^{2} \) |
| 79 | \( 1 - 9.46T + 79T^{2} \) |
| 83 | \( 1 - 4.74T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 + 2.32T + 97T^{2} \) |
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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
−8.381881095083184279177820710325, −7.73758736317545257458129944598, −7.18094939751551203611533888603, −6.52646846355498228131687416111, −5.43040463696566194565675195006, −4.57451487414564832292328231654, −3.66443910634016637423376435956, −2.36471823807202633142799799325, −1.57097174490953520786496516893, −0.788863613216677652425481328540,
0.788863613216677652425481328540, 1.57097174490953520786496516893, 2.36471823807202633142799799325, 3.66443910634016637423376435956, 4.57451487414564832292328231654, 5.43040463696566194565675195006, 6.52646846355498228131687416111, 7.18094939751551203611533888603, 7.73758736317545257458129944598, 8.381881095083184279177820710325
