L(s) = 1 | + 2.96·3-s − 4.34·5-s + 5.77·9-s − 0.777·11-s + 0.473·13-s − 12.8·15-s − 1.81·17-s − 6.26·19-s + 5.33·23-s + 13.8·25-s + 8.21·27-s + 4.40·29-s + 5.82·31-s − 2.30·33-s − 8.34·37-s + 1.40·39-s − 41-s − 6.18·43-s − 25.0·45-s − 6.01·47-s − 5.36·51-s + 11.5·53-s + 3.37·55-s − 18.5·57-s + 0.805·59-s − 11.1·61-s − 2.05·65-s + ⋯ |
L(s) = 1 | + 1.70·3-s − 1.94·5-s + 1.92·9-s − 0.234·11-s + 0.131·13-s − 3.32·15-s − 0.439·17-s − 1.43·19-s + 1.11·23-s + 2.77·25-s + 1.58·27-s + 0.818·29-s + 1.04·31-s − 0.400·33-s − 1.37·37-s + 0.224·39-s − 0.156·41-s − 0.942·43-s − 3.73·45-s − 0.877·47-s − 0.751·51-s + 1.59·53-s + 0.455·55-s − 2.45·57-s + 0.104·59-s − 1.42·61-s − 0.255·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 + 4.34T + 5T^{2} \) |
| 11 | \( 1 + 0.777T + 11T^{2} \) |
| 13 | \( 1 - 0.473T + 13T^{2} \) |
| 17 | \( 1 + 1.81T + 17T^{2} \) |
| 19 | \( 1 + 6.26T + 19T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 - 4.40T + 29T^{2} \) |
| 31 | \( 1 - 5.82T + 31T^{2} \) |
| 37 | \( 1 + 8.34T + 37T^{2} \) |
| 43 | \( 1 + 6.18T + 43T^{2} \) |
| 47 | \( 1 + 6.01T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 0.805T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 1.94T + 71T^{2} \) |
| 73 | \( 1 + 0.839T + 73T^{2} \) |
| 79 | \( 1 - 7.38T + 79T^{2} \) |
| 83 | \( 1 + 9.38T + 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.68937008234081978761689966777, −7.00792266415140188614237628528, −6.56455505673705873558421202405, −4.95150983046908999082366865458, −4.38933558324839690284552645697, −3.81125790487594468948399496446, −3.11055250262026344627040535938, −2.58614267779015558579085894071, −1.37130063040969910353596442411, 0,
1.37130063040969910353596442411, 2.58614267779015558579085894071, 3.11055250262026344627040535938, 3.81125790487594468948399496446, 4.38933558324839690284552645697, 4.95150983046908999082366865458, 6.56455505673705873558421202405, 7.00792266415140188614237628528, 7.68937008234081978761689966777