L(s) = 1 | + 2-s + 4-s + 0.316·5-s − 2.35·7-s + 8-s + 0.316·10-s − 1.38·11-s − 5.41·13-s − 2.35·14-s + 16-s + 4.25·17-s − 4.17·19-s + 0.316·20-s − 1.38·22-s − 6.49·23-s − 4.89·25-s − 5.41·26-s − 2.35·28-s + 0.574·29-s + 5.76·31-s + 32-s + 4.25·34-s − 0.745·35-s − 0.814·37-s − 4.17·38-s + 0.316·40-s + 8.31·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.141·5-s − 0.891·7-s + 0.353·8-s + 0.100·10-s − 0.418·11-s − 1.50·13-s − 0.630·14-s + 0.250·16-s + 1.03·17-s − 0.958·19-s + 0.0707·20-s − 0.295·22-s − 1.35·23-s − 0.979·25-s − 1.06·26-s − 0.445·28-s + 0.106·29-s + 1.03·31-s + 0.176·32-s + 0.729·34-s − 0.126·35-s − 0.133·37-s − 0.677·38-s + 0.0500·40-s + 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.266637274\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.266637274\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 - 0.316T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 0.574T + 29T^{2} \) |
| 31 | \( 1 - 5.76T + 31T^{2} \) |
| 37 | \( 1 + 0.814T + 37T^{2} \) |
| 41 | \( 1 - 8.31T + 41T^{2} \) |
| 43 | \( 1 - 3.92T + 43T^{2} \) |
| 47 | \( 1 - 4.19T + 47T^{2} \) |
| 53 | \( 1 - 7.98T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 9.98T + 73T^{2} \) |
| 79 | \( 1 - 4.84T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 19.0T + 97T^{2} \) |
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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
−7.86372297624715006784940059561, −6.94599673656030546785427563489, −6.43095773499847219322859263333, −5.63146901621714325277631443760, −5.17889204502957545932787460487, −4.13256443003003150671114593009, −3.69830811459657102076371782326, −2.46312805859136302228475680279, −2.32073593062530683625336198574, −0.62987201244312977935685041710,
0.62987201244312977935685041710, 2.32073593062530683625336198574, 2.46312805859136302228475680279, 3.69830811459657102076371782326, 4.13256443003003150671114593009, 5.17889204502957545932787460487, 5.63146901621714325277631443760, 6.43095773499847219322859263333, 6.94599673656030546785427563489, 7.86372297624715006784940059561