| L(s) = 1 | + 3.19·3-s − 5-s + 0.749·7-s + 7.23·9-s + 2.24·11-s + 4.35·13-s − 3.19·15-s − 0.920·17-s − 2.40·19-s + 2.39·21-s + 0.595·23-s + 25-s + 13.5·27-s + 3.83·29-s + 7.58·31-s + 7.18·33-s − 0.749·35-s + 2.30·37-s + 13.9·39-s − 6.13·41-s + 3.77·43-s − 7.23·45-s − 8.45·47-s − 6.43·49-s − 2.94·51-s + 0.775·53-s − 2.24·55-s + ⋯ |
| L(s) = 1 | + 1.84·3-s − 0.447·5-s + 0.283·7-s + 2.41·9-s + 0.677·11-s + 1.20·13-s − 0.826·15-s − 0.223·17-s − 0.550·19-s + 0.523·21-s + 0.124·23-s + 0.200·25-s + 2.60·27-s + 0.712·29-s + 1.36·31-s + 1.25·33-s − 0.126·35-s + 0.379·37-s + 2.23·39-s − 0.957·41-s + 0.576·43-s − 1.07·45-s − 1.23·47-s − 0.919·49-s − 0.412·51-s + 0.106·53-s − 0.302·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.613211019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.613211019\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 3.19T + 3T^{2} \) |
| 7 | \( 1 - 0.749T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 - 4.35T + 13T^{2} \) |
| 17 | \( 1 + 0.920T + 17T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 23 | \( 1 - 0.595T + 23T^{2} \) |
| 29 | \( 1 - 3.83T + 29T^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 - 2.30T + 37T^{2} \) |
| 41 | \( 1 + 6.13T + 41T^{2} \) |
| 43 | \( 1 - 3.77T + 43T^{2} \) |
| 47 | \( 1 + 8.45T + 47T^{2} \) |
| 53 | \( 1 - 0.775T + 53T^{2} \) |
| 59 | \( 1 + 8.45T + 59T^{2} \) |
| 61 | \( 1 + 6.99T + 61T^{2} \) |
| 67 | \( 1 + 9.96T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 3.83T + 73T^{2} \) |
| 79 | \( 1 + 0.243T + 79T^{2} \) |
| 83 | \( 1 - 7.96T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 4.60T + 97T^{2} \) |
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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
−8.075747851797705182126195009757, −7.76719894043212846337784859282, −6.67536587405279835752632184367, −6.32673175341026236002222206529, −4.79149641007017275632167535702, −4.25564048901726240990294412259, −3.49904440542846082222891742671, −2.95115429520059615753452269626, −1.92552010021362747427769266248, −1.14040302413683845109450887582,
1.14040302413683845109450887582, 1.92552010021362747427769266248, 2.95115429520059615753452269626, 3.49904440542846082222891742671, 4.25564048901726240990294412259, 4.79149641007017275632167535702, 6.32673175341026236002222206529, 6.67536587405279835752632184367, 7.76719894043212846337784859282, 8.075747851797705182126195009757
