L(s) = 1 | − 3.07·3-s + 2.30·5-s + 0.712·7-s + 6.48·9-s − 1.51·11-s − 0.395·13-s − 7.08·15-s − 0.429·17-s + 3.54·19-s − 2.19·21-s − 2.70·23-s + 0.299·25-s − 10.7·27-s + 0.745·29-s − 6.64·31-s + 4.65·33-s + 1.63·35-s + 3.72·37-s + 1.21·39-s − 9.89·41-s − 2.82·43-s + 14.9·45-s + 3.59·47-s − 6.49·49-s + 1.32·51-s − 1.92·53-s − 3.47·55-s + ⋯ |
L(s) = 1 | − 1.77·3-s + 1.02·5-s + 0.269·7-s + 2.16·9-s − 0.455·11-s − 0.109·13-s − 1.83·15-s − 0.104·17-s + 0.814·19-s − 0.478·21-s − 0.563·23-s + 0.0598·25-s − 2.06·27-s + 0.138·29-s − 1.19·31-s + 0.810·33-s + 0.277·35-s + 0.612·37-s + 0.195·39-s − 1.54·41-s − 0.430·43-s + 2.22·45-s + 0.524·47-s − 0.927·49-s + 0.185·51-s − 0.264·53-s − 0.469·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 7 | \( 1 - 0.712T + 7T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 + 0.395T + 13T^{2} \) |
| 17 | \( 1 + 0.429T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 + 2.70T + 23T^{2} \) |
| 29 | \( 1 - 0.745T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 - 3.72T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 + 1.92T + 53T^{2} \) |
| 59 | \( 1 + 1.45T + 59T^{2} \) |
| 61 | \( 1 - 1.00T + 61T^{2} \) |
| 67 | \( 1 + 2.83T + 67T^{2} \) |
| 71 | \( 1 + 5.09T + 71T^{2} \) |
| 73 | \( 1 + 3.35T + 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 5.06T + 89T^{2} \) |
| 97 | \( 1 - 3.13T + 97T^{2} \) |
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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
−7.84779055618225338480501608140, −7.16972964576164032463409915342, −6.32990945622672505417980699379, −5.86216683541404148462415554157, −5.17613357469106661406379478331, −4.73229706965483246455712993719, −3.54518213860858286008835897737, −2.13715427569300537192239170735, −1.29975186323041556279854385929, 0,
1.29975186323041556279854385929, 2.13715427569300537192239170735, 3.54518213860858286008835897737, 4.73229706965483246455712993719, 5.17613357469106661406379478331, 5.86216683541404148462415554157, 6.32990945622672505417980699379, 7.16972964576164032463409915342, 7.84779055618225338480501608140