| L(s) = 1 | + 1.88·2-s − 2.33·3-s + 1.54·4-s − 5-s − 4.39·6-s − 0.849·8-s + 2.45·9-s − 1.88·10-s − 11-s − 3.61·12-s + 4.94·13-s + 2.33·15-s − 4.69·16-s − 0.334·17-s + 4.61·18-s − 3.51·19-s − 1.54·20-s − 1.88·22-s − 4.50·23-s + 1.98·24-s + 25-s + 9.32·26-s + 1.28·27-s + 8.56·29-s + 4.39·30-s − 6.65·31-s − 7.15·32-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 1.34·3-s + 0.774·4-s − 0.447·5-s − 1.79·6-s − 0.300·8-s + 0.816·9-s − 0.595·10-s − 0.301·11-s − 1.04·12-s + 1.37·13-s + 0.602·15-s − 1.17·16-s − 0.0811·17-s + 1.08·18-s − 0.806·19-s − 0.346·20-s − 0.401·22-s − 0.938·23-s + 0.404·24-s + 0.200·25-s + 1.82·26-s + 0.246·27-s + 1.59·29-s + 0.803·30-s − 1.19·31-s − 1.26·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664352558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664352558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 3 | \( 1 + 2.33T + 3T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 + 0.334T + 17T^{2} \) |
| 19 | \( 1 + 3.51T + 19T^{2} \) |
| 23 | \( 1 + 4.50T + 23T^{2} \) |
| 29 | \( 1 - 8.56T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 - 7.55T + 41T^{2} \) |
| 43 | \( 1 + 1.93T + 43T^{2} \) |
| 47 | \( 1 + 8.75T + 47T^{2} \) |
| 53 | \( 1 - 8.86T + 53T^{2} \) |
| 59 | \( 1 - 8.11T + 59T^{2} \) |
| 61 | \( 1 - 6.39T + 61T^{2} \) |
| 67 | \( 1 - 7.36T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 8.18T + 83T^{2} \) |
| 89 | \( 1 - 7.28T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.680733612577573553376365760642, −8.025622417679414101087659143532, −6.68923517935486212656346762271, −6.39365506312230578354483258028, −5.62410995893615670210881692643, −5.01532612475959465009948621809, −4.16636694534570710524791161641, −3.60622906175484162158823193116, −2.34118642574917535916329945829, −0.69274274210182360683980271693,
0.69274274210182360683980271693, 2.34118642574917535916329945829, 3.60622906175484162158823193116, 4.16636694534570710524791161641, 5.01532612475959465009948621809, 5.62410995893615670210881692643, 6.39365506312230578354483258028, 6.68923517935486212656346762271, 8.025622417679414101087659143532, 8.680733612577573553376365760642
