| L(s) = 1 | − 0.195·2-s − 0.259·3-s − 1.96·4-s + 3.93·5-s + 0.0508·6-s + 0.775·8-s − 2.93·9-s − 0.769·10-s + 4.50·11-s + 0.509·12-s + 13-s − 1.02·15-s + 3.77·16-s − 2.28·17-s + 0.573·18-s − 1.78·19-s − 7.71·20-s − 0.881·22-s + 1.74·23-s − 0.201·24-s + 10.4·25-s − 0.195·26-s + 1.54·27-s + 1.65·29-s + 0.199·30-s + 5.60·31-s − 2.28·32-s + ⋯ |
| L(s) = 1 | − 0.138·2-s − 0.149·3-s − 0.980·4-s + 1.75·5-s + 0.0207·6-s + 0.274·8-s − 0.977·9-s − 0.243·10-s + 1.35·11-s + 0.147·12-s + 0.277·13-s − 0.263·15-s + 0.942·16-s − 0.553·17-s + 0.135·18-s − 0.410·19-s − 1.72·20-s − 0.187·22-s + 0.362·23-s − 0.0411·24-s + 2.09·25-s − 0.0383·26-s + 0.296·27-s + 0.306·29-s + 0.0364·30-s + 1.00·31-s − 0.404·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.457931517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.457931517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.195T + 2T^{2} \) |
| 3 | \( 1 + 0.259T + 3T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 11 | \( 1 - 4.50T + 11T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 + 1.78T + 19T^{2} \) |
| 23 | \( 1 - 1.74T + 23T^{2} \) |
| 29 | \( 1 - 1.65T + 29T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 - 7.14T + 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 - 6.81T + 43T^{2} \) |
| 47 | \( 1 - 3.54T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 - 7.54T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 9.54T + 71T^{2} \) |
| 73 | \( 1 - 1.08T + 73T^{2} \) |
| 79 | \( 1 - 0.791T + 79T^{2} \) |
| 83 | \( 1 + 7.14T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 8.81T + 97T^{2} \) |
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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
−10.36110620479067596003155069244, −9.572817839349911366904369790832, −8.975133828205401973565169243250, −8.392307526431928522157372431685, −6.69757501810445103551369381232, −6.02883304303346113051358692948, −5.22509066933486439733252162204, −4.12836263864436883298067059119, −2.63643294720511834793319575028, −1.19271414644531081978401165037,
1.19271414644531081978401165037, 2.63643294720511834793319575028, 4.12836263864436883298067059119, 5.22509066933486439733252162204, 6.02883304303346113051358692948, 6.69757501810445103551369381232, 8.392307526431928522157372431685, 8.975133828205401973565169243250, 9.572817839349911366904369790832, 10.36110620479067596003155069244
