| L(s) = 1 | + 3-s + 5-s − 2.42·7-s + 9-s + 2.63·11-s + 3.28·13-s + 15-s + 8.05·17-s − 1.54·19-s − 2.42·21-s + 1.83·23-s + 25-s + 27-s − 6.09·29-s + 5.70·31-s + 2.63·33-s − 2.42·35-s − 5.18·37-s + 3.28·39-s + 0.845·41-s − 1.75·43-s + 45-s + 9.75·47-s − 1.11·49-s + 8.05·51-s + 1.12·53-s + 2.63·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.917·7-s + 0.333·9-s + 0.792·11-s + 0.911·13-s + 0.258·15-s + 1.95·17-s − 0.354·19-s − 0.529·21-s + 0.382·23-s + 0.200·25-s + 0.192·27-s − 1.13·29-s + 1.02·31-s + 0.457·33-s − 0.410·35-s − 0.851·37-s + 0.526·39-s + 0.131·41-s − 0.267·43-s + 0.149·45-s + 1.42·47-s − 0.158·49-s + 1.12·51-s + 0.154·53-s + 0.354·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.105656209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.105656209\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 2.42T + 7T^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 - 8.05T + 17T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 23 | \( 1 - 1.83T + 23T^{2} \) |
| 29 | \( 1 + 6.09T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 + 5.18T + 37T^{2} \) |
| 41 | \( 1 - 0.845T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 - 9.75T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 - 0.551T + 59T^{2} \) |
| 61 | \( 1 + 5.56T + 61T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 - 4.42T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 1.04T + 97T^{2} \) |
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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
−7.84578409928363676599700467678, −7.13549228978471244231328140357, −6.40093373675074628398865373198, −5.89737029943140366717175136348, −5.12748668556653681262021443162, −3.98051456506430132190873111193, −3.49590961569892297952179998514, −2.83613994195869599740054970425, −1.72261113257134168632192950094, −0.899649525343639138748590313804,
0.899649525343639138748590313804, 1.72261113257134168632192950094, 2.83613994195869599740054970425, 3.49590961569892297952179998514, 3.98051456506430132190873111193, 5.12748668556653681262021443162, 5.89737029943140366717175136348, 6.40093373675074628398865373198, 7.13549228978471244231328140357, 7.84578409928363676599700467678
