| L(s) = 1 | + 0.772·2-s − 3-s − 1.40·4-s + 5-s − 0.772·6-s + 2.17·7-s − 2.62·8-s + 9-s + 0.772·10-s + 3·11-s + 1.40·12-s − 0.629·13-s + 1.68·14-s − 15-s + 0.772·16-s + 4.17·17-s + 0.772·18-s + 4.80·19-s − 1.40·20-s − 2.17·21-s + 2.31·22-s + 2.08·23-s + 2.62·24-s + 25-s − 0.486·26-s − 27-s − 3.05·28-s + ⋯ |
| L(s) = 1 | + 0.546·2-s − 0.577·3-s − 0.701·4-s + 0.447·5-s − 0.315·6-s + 0.822·7-s − 0.929·8-s + 0.333·9-s + 0.244·10-s + 0.904·11-s + 0.404·12-s − 0.174·13-s + 0.449·14-s − 0.258·15-s + 0.193·16-s + 1.01·17-s + 0.182·18-s + 1.10·19-s − 0.313·20-s − 0.474·21-s + 0.494·22-s + 0.434·23-s + 0.536·24-s + 0.200·25-s − 0.0954·26-s − 0.192·27-s − 0.576·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.539770313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539770313\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.772T + 2T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 0.629T + 13T^{2} \) |
| 17 | \( 1 - 4.17T + 17T^{2} \) |
| 19 | \( 1 - 4.80T + 19T^{2} \) |
| 23 | \( 1 - 2.08T + 23T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.08T + 37T^{2} \) |
| 41 | \( 1 - 0.824T + 41T^{2} \) |
| 43 | \( 1 + 8.72T + 43T^{2} \) |
| 47 | \( 1 + 8.98T + 47T^{2} \) |
| 53 | \( 1 - 6.88T + 53T^{2} \) |
| 59 | \( 1 - 6.45T + 59T^{2} \) |
| 61 | \( 1 + 2.80T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 2.63T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 7.62T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 - 0.538T + 97T^{2} \) |
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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
−11.50639152931305203063062394402, −10.14111263568956731782780792449, −9.477996869809564280760342564614, −8.480133861513070054844477896308, −7.35637204840578365156498073296, −6.10972133769248522691221431287, −5.29995804282330969784902939486, −4.53167440669236326129400406824, −3.29437045818504435624558241890, −1.27925173991626807871074967574,
1.27925173991626807871074967574, 3.29437045818504435624558241890, 4.53167440669236326129400406824, 5.29995804282330969784902939486, 6.10972133769248522691221431287, 7.35637204840578365156498073296, 8.480133861513070054844477896308, 9.477996869809564280760342564614, 10.14111263568956731782780792449, 11.50639152931305203063062394402
