| L(s) = 1 | + 1.83·2-s + 0.853·3-s + 1.35·4-s + 2.62·5-s + 1.56·6-s − 1.17·8-s − 2.27·9-s + 4.81·10-s + 3.26·11-s + 1.15·12-s + 13-s + 2.24·15-s − 4.87·16-s + 4.53·17-s − 4.16·18-s + 4.06·19-s + 3.56·20-s + 5.98·22-s − 4.53·23-s − 1.00·24-s + 1.89·25-s + 1.83·26-s − 4.50·27-s − 1.42·29-s + 4.10·30-s − 2.80·31-s − 6.57·32-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.492·3-s + 0.678·4-s + 1.17·5-s + 0.638·6-s − 0.416·8-s − 0.757·9-s + 1.52·10-s + 0.984·11-s + 0.334·12-s + 0.277·13-s + 0.578·15-s − 1.21·16-s + 1.09·17-s − 0.980·18-s + 0.932·19-s + 0.797·20-s + 1.27·22-s − 0.945·23-s − 0.205·24-s + 0.378·25-s + 0.359·26-s − 0.866·27-s − 0.264·29-s + 0.749·30-s − 0.503·31-s − 1.16·32-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
| \(L(1)\) |
\(\approx\) |
\(3.658519729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.658519729\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 - 0.853T + 3T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 23 | \( 1 + 4.53T + 23T^{2} \) |
| 29 | \( 1 + 1.42T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 2.84T + 41T^{2} \) |
| 43 | \( 1 - 9.72T + 43T^{2} \) |
| 47 | \( 1 + 9.44T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 1.98T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−10.66354676703402963636632321490, −9.486718033023664631628693345734, −9.120832731181008022570243555899, −7.910692728784159615642134206833, −6.59152220820597614388635124341, −5.79198189725718630430871254835, −5.26562698114258209798116913137, −3.83667107048168242498577795091, −3.09389315517282991663458065506, −1.82639530032567334447259075068,
1.82639530032567334447259075068, 3.09389315517282991663458065506, 3.83667107048168242498577795091, 5.26562698114258209798116913137, 5.79198189725718630430871254835, 6.59152220820597614388635124341, 7.910692728784159615642134206833, 9.120832731181008022570243555899, 9.486718033023664631628693345734, 10.66354676703402963636632321490
