| L(s) = 1 | + 0.646·2-s − 3.35·3-s − 1.58·4-s + 1.94·5-s − 2.17·6-s − 2.31·8-s + 8.28·9-s + 1.26·10-s − 1.14·11-s + 5.31·12-s − 1.35·13-s − 6.54·15-s + 1.66·16-s − 0.375·17-s + 5.35·18-s − 2.83·19-s − 3.08·20-s − 0.740·22-s + 1.07·23-s + 7.77·24-s − 1.20·25-s − 0.877·26-s − 17.7·27-s + 5.91·29-s − 4.23·30-s − 5.79·31-s + 5.70·32-s + ⋯ |
| L(s) = 1 | + 0.457·2-s − 1.93·3-s − 0.790·4-s + 0.871·5-s − 0.886·6-s − 0.818·8-s + 2.76·9-s + 0.398·10-s − 0.345·11-s + 1.53·12-s − 0.376·13-s − 1.69·15-s + 0.416·16-s − 0.0911·17-s + 1.26·18-s − 0.649·19-s − 0.689·20-s − 0.157·22-s + 0.224·23-s + 1.58·24-s − 0.240·25-s − 0.172·26-s − 3.41·27-s + 1.09·29-s − 0.772·30-s − 1.04·31-s + 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 0.646T + 2T^{2} \) |
| 3 | \( 1 + 3.35T + 3T^{2} \) |
| 5 | \( 1 - 1.94T + 5T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 + 0.375T + 17T^{2} \) |
| 19 | \( 1 + 2.83T + 19T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 - 5.91T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 - 0.731T + 37T^{2} \) |
| 41 | \( 1 - 8.15T + 41T^{2} \) |
| 43 | \( 1 + 1.00T + 43T^{2} \) |
| 47 | \( 1 - 7.36T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 2.02T + 61T^{2} \) |
| 67 | \( 1 - 7.14T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 9.77T + 83T^{2} \) |
| 89 | \( 1 + 8.39T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.108376185046105104976508971749, −7.08654909345578618290723582330, −6.41104530019400024081415157965, −5.62354178983600802489161399431, −5.42158229567536551979924036020, −4.53897876684820166839461184155, −3.96852184852332409038623723026, −2.41980985647079746591033209491, −1.11757741760340830825559785151, 0,
1.11757741760340830825559785151, 2.41980985647079746591033209491, 3.96852184852332409038623723026, 4.53897876684820166839461184155, 5.42158229567536551979924036020, 5.62354178983600802489161399431, 6.41104530019400024081415157965, 7.08654909345578618290723582330, 8.108376185046105104976508971749
