| L(s) = 1 | − 1.17·2-s + 3.35·3-s − 0.616·4-s + 3.14·5-s − 3.94·6-s + 3.07·8-s + 8.22·9-s − 3.70·10-s − 0.773·11-s − 2.06·12-s + 13-s + 10.5·15-s − 2.38·16-s − 5.75·17-s − 9.67·18-s − 1.22·19-s − 1.94·20-s + 0.909·22-s − 2.99·23-s + 10.3·24-s + 4.91·25-s − 1.17·26-s + 17.5·27-s − 2.46·29-s − 12.4·30-s − 6.13·31-s − 3.34·32-s + ⋯ |
| L(s) = 1 | − 0.831·2-s + 1.93·3-s − 0.308·4-s + 1.40·5-s − 1.60·6-s + 1.08·8-s + 2.74·9-s − 1.17·10-s − 0.233·11-s − 0.596·12-s + 0.277·13-s + 2.72·15-s − 0.596·16-s − 1.39·17-s − 2.28·18-s − 0.280·19-s − 0.434·20-s + 0.193·22-s − 0.623·23-s + 2.10·24-s + 0.982·25-s − 0.230·26-s + 3.37·27-s − 0.458·29-s − 2.26·30-s − 1.10·31-s − 0.591·32-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
| \(L(1)\) |
\(\approx\) |
\(2.081214073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.081214073\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.17T + 2T^{2} \) |
| 3 | \( 1 - 3.35T + 3T^{2} \) |
| 5 | \( 1 - 3.14T + 5T^{2} \) |
| 11 | \( 1 + 0.773T + 11T^{2} \) |
| 17 | \( 1 + 5.75T + 17T^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 23 | \( 1 + 2.99T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 37 | \( 1 - 4.99T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 + 9.79T + 53T^{2} \) |
| 59 | \( 1 - 2.50T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 5.17T + 73T^{2} \) |
| 79 | \( 1 + 0.542T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 9.23T + 89T^{2} \) |
| 97 | \( 1 + 1.26T + 97T^{2} \) |
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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
−10.13922704394775975336900034236, −9.331543853372087150861870829455, −9.091908285696682337428165960129, −8.252003237231177130657331323688, −7.46360338335008311325997787584, −6.38482276822183894928856597019, −4.84002741993096800399995300541, −3.78307621201181422753251217693, −2.35237178692968263429215304691, −1.68756983034729400615022091457,
1.68756983034729400615022091457, 2.35237178692968263429215304691, 3.78307621201181422753251217693, 4.84002741993096800399995300541, 6.38482276822183894928856597019, 7.46360338335008311325997787584, 8.252003237231177130657331323688, 9.091908285696682337428165960129, 9.331543853372087150861870829455, 10.13922704394775975336900034236
