| L(s) = 1 | − 2.93·3-s + 1.25·5-s − 7-s + 5.61·9-s − 2.50·11-s − 3.68·15-s − 17-s + 3.87·19-s + 2.93·21-s + 4.37·23-s − 3.42·25-s − 7.68·27-s − 7.87·29-s − 1.25·31-s + 7.36·33-s − 1.25·35-s + 3.87·37-s + 9.28·41-s + 1.41·43-s + 7.04·45-s + 7.74·47-s + 49-s + 2.93·51-s + 1.38·53-s − 3.14·55-s − 11.3·57-s − 8.34·59-s + ⋯ |
| L(s) = 1 | − 1.69·3-s + 0.560·5-s − 0.377·7-s + 1.87·9-s − 0.756·11-s − 0.950·15-s − 0.242·17-s + 0.888·19-s + 0.640·21-s + 0.913·23-s − 0.685·25-s − 1.47·27-s − 1.46·29-s − 0.225·31-s + 1.28·33-s − 0.211·35-s + 0.636·37-s + 1.44·41-s + 0.215·43-s + 1.05·45-s + 1.12·47-s + 0.142·49-s + 0.411·51-s + 0.190·53-s − 0.424·55-s − 1.50·57-s − 1.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 5 | \( 1 - 1.25T + 5T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 - 3.87T + 37T^{2} \) |
| 41 | \( 1 - 9.28T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 - 7.74T + 47T^{2} \) |
| 53 | \( 1 - 1.38T + 53T^{2} \) |
| 59 | \( 1 + 8.34T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 0.475T + 71T^{2} \) |
| 73 | \( 1 - 0.427T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 1.83T + 83T^{2} \) |
| 89 | \( 1 + 2.37T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.198250864702968111145899069810, −7.67612604488545837635398672755, −7.17938206566582065798415108528, −6.10351554236201601384242826197, −5.72990585160382804511369732012, −5.02525681740044008423686220107, −4.07305135997770552617002934518, −2.70552934607491041694729981692, −1.32108607976346260126523355971, 0,
1.32108607976346260126523355971, 2.70552934607491041694729981692, 4.07305135997770552617002934518, 5.02525681740044008423686220107, 5.72990585160382804511369732012, 6.10351554236201601384242826197, 7.17938206566582065798415108528, 7.67612604488545837635398672755, 9.198250864702968111145899069810
