| L(s) = 1 | − 3-s − 5-s + 4.46·7-s + 9-s + 3.39·11-s − 4.70·13-s + 15-s − 6.54·17-s − 6.22·19-s − 4.46·21-s + 23-s + 25-s − 27-s + 0.448·29-s − 5.15·31-s − 3.39·33-s − 4.46·35-s − 5.98·37-s + 4.70·39-s + 3.07·41-s − 4.47·43-s − 45-s + 6.22·47-s + 12.9·49-s + 6.54·51-s − 2.59·53-s − 3.39·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.68·7-s + 0.333·9-s + 1.02·11-s − 1.30·13-s + 0.258·15-s − 1.58·17-s − 1.42·19-s − 0.974·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.0832·29-s − 0.925·31-s − 0.590·33-s − 0.754·35-s − 0.984·37-s + 0.753·39-s + 0.479·41-s − 0.683·43-s − 0.149·45-s + 0.908·47-s + 1.84·49-s + 0.916·51-s − 0.355·53-s − 0.457·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 19 | \( 1 + 6.22T + 19T^{2} \) |
| 29 | \( 1 - 0.448T + 29T^{2} \) |
| 31 | \( 1 + 5.15T + 31T^{2} \) |
| 37 | \( 1 + 5.98T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 + 2.59T + 53T^{2} \) |
| 59 | \( 1 + 1.55T + 59T^{2} \) |
| 61 | \( 1 + 4.91T + 61T^{2} \) |
| 67 | \( 1 - 6.08T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + 8.70T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 4.30T + 97T^{2} \) |
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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
−8.612287802091070089948138424637, −7.52880888997681561854552068597, −7.03033377115956082779424559306, −6.18554474053802302852025575365, −5.09098790663885614863503406183, −4.55180035448100883605421666169, −3.99224650779977918821725671012, −2.34458977486626211746305060891, −1.56529592758404973736493147687, 0,
1.56529592758404973736493147687, 2.34458977486626211746305060891, 3.99224650779977918821725671012, 4.55180035448100883605421666169, 5.09098790663885614863503406183, 6.18554474053802302852025575365, 7.03033377115956082779424559306, 7.52880888997681561854552068597, 8.612287802091070089948138424637
