L(s) = 1 | − 0.810·2-s + 3-s − 1.34·4-s + 0.143·5-s − 0.810·6-s − 7-s + 2.70·8-s + 9-s − 0.116·10-s − 0.888·11-s − 1.34·12-s − 0.424·13-s + 0.810·14-s + 0.143·15-s + 0.489·16-s − 0.886·17-s − 0.810·18-s − 8.58·19-s − 0.192·20-s − 21-s + 0.720·22-s + 4.66·23-s + 2.70·24-s − 4.97·25-s + 0.343·26-s + 27-s + 1.34·28-s + ⋯ |
L(s) = 1 | − 0.573·2-s + 0.577·3-s − 0.671·4-s + 0.0641·5-s − 0.330·6-s − 0.377·7-s + 0.958·8-s + 0.333·9-s − 0.0367·10-s − 0.268·11-s − 0.387·12-s − 0.117·13-s + 0.216·14-s + 0.0370·15-s + 0.122·16-s − 0.215·17-s − 0.191·18-s − 1.96·19-s − 0.0430·20-s − 0.218·21-s + 0.153·22-s + 0.972·23-s + 0.553·24-s − 0.995·25-s + 0.0674·26-s + 0.192·27-s + 0.253·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 + 0.810T + 2T^{2} \) |
| 5 | \( 1 - 0.143T + 5T^{2} \) |
| 11 | \( 1 + 0.888T + 11T^{2} \) |
| 13 | \( 1 + 0.424T + 13T^{2} \) |
| 17 | \( 1 + 0.886T + 17T^{2} \) |
| 19 | \( 1 + 8.58T + 19T^{2} \) |
| 23 | \( 1 - 4.66T + 23T^{2} \) |
| 29 | \( 1 - 7.67T + 29T^{2} \) |
| 31 | \( 1 - 5.71T + 31T^{2} \) |
| 37 | \( 1 + 3.86T + 37T^{2} \) |
| 41 | \( 1 - 3.14T + 41T^{2} \) |
| 43 | \( 1 - 4.97T + 43T^{2} \) |
| 47 | \( 1 - 8.47T + 47T^{2} \) |
| 53 | \( 1 - 0.270T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 8.57T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 3.15T + 73T^{2} \) |
| 79 | \( 1 + 8.70T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 + 8.01T + 97T^{2} \) |
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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
−7.66350628311731383715340063851, −6.94406341733516513615195326557, −6.23633786117511106755277253226, −5.34911573578816955788403347088, −4.35955379156130775507539049783, −4.14638570413755308606637623780, −2.95037129591724616513847189824, −2.24621733638011571774222328372, −1.12958238908830229180383132468, 0,
1.12958238908830229180383132468, 2.24621733638011571774222328372, 2.95037129591724616513847189824, 4.14638570413755308606637623780, 4.35955379156130775507539049783, 5.34911573578816955788403347088, 6.23633786117511106755277253226, 6.94406341733516513615195326557, 7.66350628311731383715340063851