L(s) = 1 | − 3.11·3-s − 5-s + 7-s + 6.67·9-s + 4.70·11-s + 2.64·13-s + 3.11·15-s − 0.482·17-s − 6.87·19-s − 3.11·21-s − 23-s + 25-s − 11.4·27-s + 4.63·29-s − 2.89·31-s − 14.6·33-s − 35-s − 9.50·37-s − 8.22·39-s − 5.73·41-s + 5.24·43-s − 6.67·45-s − 8.14·47-s + 49-s + 1.50·51-s + 0.237·53-s − 4.70·55-s + ⋯ |
L(s) = 1 | − 1.79·3-s − 0.447·5-s + 0.377·7-s + 2.22·9-s + 1.41·11-s + 0.733·13-s + 0.803·15-s − 0.117·17-s − 1.57·19-s − 0.678·21-s − 0.208·23-s + 0.200·25-s − 2.19·27-s + 0.860·29-s − 0.520·31-s − 2.54·33-s − 0.169·35-s − 1.56·37-s − 1.31·39-s − 0.894·41-s + 0.799·43-s − 0.994·45-s − 1.18·47-s + 0.142·49-s + 0.210·51-s + 0.0326·53-s − 0.634·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 + 0.482T + 17T^{2} \) |
| 19 | \( 1 + 6.87T + 19T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 + 9.50T + 37T^{2} \) |
| 41 | \( 1 + 5.73T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 - 0.237T + 53T^{2} \) |
| 59 | \( 1 - 8.37T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 5.46T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 8.15T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 5.91T + 89T^{2} \) |
| 97 | \( 1 - 9.00T + 97T^{2} \) |
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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
−7.35886693753428360677191303310, −6.71892862054624583421887571278, −6.32656230675028193178431115667, −5.62905927160912363778408696230, −4.77585300992958733531541516202, −4.21478275097699775359479742742, −3.58151669421006358468956890027, −1.88851513152694080186761653130, −1.11398194510814839328259518590, 0,
1.11398194510814839328259518590, 1.88851513152694080186761653130, 3.58151669421006358468956890027, 4.21478275097699775359479742742, 4.77585300992958733531541516202, 5.62905927160912363778408696230, 6.32656230675028193178431115667, 6.71892862054624583421887571278, 7.35886693753428360677191303310