| L(s) = 1 | − 3-s − 5-s − 3·7-s + 9-s + 2.31·11-s + 4.31·13-s + 15-s + 3.31·17-s − 8.31·19-s + 3·21-s + 23-s + 25-s − 27-s − 7.31·29-s + 3.63·31-s − 2.31·33-s + 3·35-s − 1.63·37-s − 4.31·39-s − 3.31·41-s − 10.6·43-s − 45-s − 8.94·47-s + 2·49-s − 3.31·51-s + 9.94·53-s − 2.31·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 0.333·9-s + 0.698·11-s + 1.19·13-s + 0.258·15-s + 0.804·17-s − 1.90·19-s + 0.654·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 1.35·29-s + 0.652·31-s − 0.403·33-s + 0.507·35-s − 0.268·37-s − 0.691·39-s − 0.517·41-s − 1.62·43-s − 0.149·45-s − 1.30·47-s + 0.285·49-s − 0.464·51-s + 1.36·53-s − 0.312·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.042218336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.042218336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 2.31T + 11T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 19 | \( 1 + 8.31T + 19T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 - 3.63T + 31T^{2} \) |
| 37 | \( 1 + 1.63T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 - 7.94T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 8.31T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 1.31T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.664053628097487819825069403003, −8.260380654350630025688520560149, −6.97760543701485260247381425046, −6.55566259162270283067746650335, −5.92732978606312590007903538996, −4.95053186537450852652801699625, −3.74432057978739411699899357295, −3.56700508338107662618926018460, −1.98986803113796878357263249133, −0.64724250132852606189154653416,
0.64724250132852606189154653416, 1.98986803113796878357263249133, 3.56700508338107662618926018460, 3.74432057978739411699899357295, 4.95053186537450852652801699625, 5.92732978606312590007903538996, 6.55566259162270283067746650335, 6.97760543701485260247381425046, 8.260380654350630025688520560149, 8.664053628097487819825069403003
