| L(s) = 1 | + 2-s − 1.37·3-s + 4-s + 0.347·5-s − 1.37·6-s − 0.477·7-s + 8-s − 1.10·9-s + 0.347·10-s + 2.58·11-s − 1.37·12-s − 4.48·13-s − 0.477·14-s − 0.477·15-s + 16-s + 2.00·17-s − 1.10·18-s + 3.52·19-s + 0.347·20-s + 0.657·21-s + 2.58·22-s + 5.20·23-s − 1.37·24-s − 4.87·25-s − 4.48·26-s + 5.65·27-s − 0.477·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.794·3-s + 0.5·4-s + 0.155·5-s − 0.561·6-s − 0.180·7-s + 0.353·8-s − 0.369·9-s + 0.109·10-s + 0.779·11-s − 0.397·12-s − 1.24·13-s − 0.127·14-s − 0.123·15-s + 0.250·16-s + 0.487·17-s − 0.261·18-s + 0.808·19-s + 0.0776·20-s + 0.143·21-s + 0.551·22-s + 1.08·23-s − 0.280·24-s − 0.975·25-s − 0.879·26-s + 1.08·27-s − 0.0902·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.060324082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.060324082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 1.37T + 3T^{2} \) |
| 5 | \( 1 - 0.347T + 5T^{2} \) |
| 7 | \( 1 + 0.477T + 7T^{2} \) |
| 11 | \( 1 - 2.58T + 11T^{2} \) |
| 13 | \( 1 + 4.48T + 13T^{2} \) |
| 17 | \( 1 - 2.00T + 17T^{2} \) |
| 19 | \( 1 - 3.52T + 19T^{2} \) |
| 23 | \( 1 - 5.20T + 23T^{2} \) |
| 29 | \( 1 - 0.228T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 8.85T + 43T^{2} \) |
| 47 | \( 1 - 7.45T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 1.43T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 8.87T + 71T^{2} \) |
| 73 | \( 1 - 7.18T + 73T^{2} \) |
| 79 | \( 1 - 3.53T + 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 - 3.82T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−9.060363905989055270305199977971, −7.70563738870974064766500227543, −7.21833227033160970081620727576, −6.31513441986480564058758682809, −5.63583451200388137751111347857, −5.11111599922477362081407936366, −4.17702763883323041864582476227, −3.21134260863947954665160422288, −2.26361945636921679297954379357, −0.836519248136080886170989545859,
0.836519248136080886170989545859, 2.26361945636921679297954379357, 3.21134260863947954665160422288, 4.17702763883323041864582476227, 5.11111599922477362081407936366, 5.63583451200388137751111347857, 6.31513441986480564058758682809, 7.21833227033160970081620727576, 7.70563738870974064766500227543, 9.060363905989055270305199977971
