| L(s) = 1 | + 0.691·2-s − 3-s − 1.52·4-s − 2.34·5-s − 0.691·6-s − 2.43·8-s + 9-s − 1.62·10-s − 2.65·11-s + 1.52·12-s + 13-s + 2.34·15-s + 1.35·16-s − 2.32·17-s + 0.691·18-s − 8.17·19-s + 3.57·20-s − 1.83·22-s − 3.45·23-s + 2.43·24-s + 0.521·25-s + 0.691·26-s − 27-s + 9.71·29-s + 1.62·30-s − 6.96·31-s + 5.81·32-s + ⋯ |
| L(s) = 1 | + 0.488·2-s − 0.577·3-s − 0.760·4-s − 1.05·5-s − 0.282·6-s − 0.861·8-s + 0.333·9-s − 0.513·10-s − 0.799·11-s + 0.439·12-s + 0.277·13-s + 0.606·15-s + 0.339·16-s − 0.563·17-s + 0.162·18-s − 1.87·19-s + 0.799·20-s − 0.391·22-s − 0.719·23-s + 0.497·24-s + 0.104·25-s + 0.135·26-s − 0.192·27-s + 1.80·29-s + 0.296·30-s − 1.25·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5965119627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5965119627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.691T + 2T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 17 | \( 1 + 2.32T + 17T^{2} \) |
| 19 | \( 1 + 8.17T + 19T^{2} \) |
| 23 | \( 1 + 3.45T + 23T^{2} \) |
| 29 | \( 1 - 9.71T + 29T^{2} \) |
| 31 | \( 1 + 6.96T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 + 9.43T + 43T^{2} \) |
| 47 | \( 1 - 2.87T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 0.732T + 59T^{2} \) |
| 61 | \( 1 - 1.33T + 61T^{2} \) |
| 67 | \( 1 + 5.73T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 - 1.49T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 8.85T + 89T^{2} \) |
| 97 | \( 1 - 7.74T + 97T^{2} \) |
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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
−9.070418451558821587593585239314, −8.323319498444916815945106702343, −7.81096040358772600314396793274, −6.62488589588552562901964204565, −5.97080414281870700675000863276, −4.95241370459303052949948059431, −4.31387578435164899827944280303, −3.71469885961035017843867849924, −2.41087663814596105739866921929, −0.47783295421864503856926331910,
0.47783295421864503856926331910, 2.41087663814596105739866921929, 3.71469885961035017843867849924, 4.31387578435164899827944280303, 4.95241370459303052949948059431, 5.97080414281870700675000863276, 6.62488589588552562901964204565, 7.81096040358772600314396793274, 8.323319498444916815945106702343, 9.070418451558821587593585239314
