| L(s) = 1 | − 3-s + 1.23·5-s − 1.23·7-s + 9-s − 2·11-s + 13-s − 1.23·15-s + 4.47·17-s − 5.23·19-s + 1.23·21-s − 2.47·23-s − 3.47·25-s − 27-s − 4.47·29-s + 7.70·31-s + 2·33-s − 1.52·35-s − 0.472·37-s − 39-s + 1.23·41-s − 6.47·43-s + 1.23·45-s + 6.94·47-s − 5.47·49-s − 4.47·51-s − 8.47·53-s − 2.47·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.552·5-s − 0.467·7-s + 0.333·9-s − 0.603·11-s + 0.277·13-s − 0.319·15-s + 1.08·17-s − 1.20·19-s + 0.269·21-s − 0.515·23-s − 0.694·25-s − 0.192·27-s − 0.830·29-s + 1.38·31-s + 0.348·33-s − 0.258·35-s − 0.0776·37-s − 0.160·39-s + 0.193·41-s − 0.986·43-s + 0.184·45-s + 1.01·47-s − 0.781·49-s − 0.626·51-s − 1.16·53-s − 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 - 0.472T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−8.446766231860281538339079837982, −7.82208154029171334108004394595, −6.83884835363265551512566659511, −6.07883335818014583897818325058, −5.61582202023066988955274558953, −4.64015006392305301207632595168, −3.69468881758289492211632868874, −2.61564350980541336147873064546, −1.51267360636805960213934668581, 0,
1.51267360636805960213934668581, 2.61564350980541336147873064546, 3.69468881758289492211632868874, 4.64015006392305301207632595168, 5.61582202023066988955274558953, 6.07883335818014583897818325058, 6.83884835363265551512566659511, 7.82208154029171334108004394595, 8.446766231860281538339079837982
