| L(s) = 1 | − 0.790·2-s − 1.37·4-s − 4.15·5-s + 7-s + 2.66·8-s + 3.28·10-s + 2.46·13-s − 0.790·14-s + 0.639·16-s − 1.99·17-s + 3.11·19-s + 5.70·20-s + 3.45·23-s + 12.2·25-s − 1.95·26-s − 1.37·28-s − 3.32·29-s − 0.288·31-s − 5.84·32-s + 1.57·34-s − 4.15·35-s + 7.59·37-s − 2.46·38-s − 11.0·40-s + 2.32·41-s − 8.35·43-s − 2.73·46-s + ⋯ |
| L(s) = 1 | − 0.559·2-s − 0.687·4-s − 1.85·5-s + 0.377·7-s + 0.943·8-s + 1.03·10-s + 0.684·13-s − 0.211·14-s + 0.159·16-s − 0.483·17-s + 0.713·19-s + 1.27·20-s + 0.721·23-s + 2.44·25-s − 0.382·26-s − 0.259·28-s − 0.617·29-s − 0.0518·31-s − 1.03·32-s + 0.270·34-s − 0.701·35-s + 1.24·37-s − 0.399·38-s − 1.75·40-s + 0.363·41-s − 1.27·43-s − 0.403·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7321515824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7321515824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.790T + 2T^{2} \) |
| 5 | \( 1 + 4.15T + 5T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 + 1.99T + 17T^{2} \) |
| 19 | \( 1 - 3.11T + 19T^{2} \) |
| 23 | \( 1 - 3.45T + 23T^{2} \) |
| 29 | \( 1 + 3.32T + 29T^{2} \) |
| 31 | \( 1 + 0.288T + 31T^{2} \) |
| 37 | \( 1 - 7.59T + 37T^{2} \) |
| 41 | \( 1 - 2.32T + 41T^{2} \) |
| 43 | \( 1 + 8.35T + 43T^{2} \) |
| 47 | \( 1 - 2.94T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 7.82T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 8.86T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 7.28T + 97T^{2} \) |
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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
−7.927294602214490653225562993741, −7.41826873990875215549900441093, −6.86703306823219126433517389391, −5.65368557038437427151555828473, −4.87282576412147777619588623939, −4.20166097491389441778645521337, −3.75825618408176339235641321904, −2.85546672509496285314497427559, −1.34667682045832619217140641807, −0.52912085240392158444916125244,
0.52912085240392158444916125244, 1.34667682045832619217140641807, 2.85546672509496285314497427559, 3.75825618408176339235641321904, 4.20166097491389441778645521337, 4.87282576412147777619588623939, 5.65368557038437427151555828473, 6.86703306823219126433517389391, 7.41826873990875215549900441093, 7.927294602214490653225562993741
