| L(s) = 1 | + 1.87·2-s + 1.52·4-s + 0.769·5-s − 0.850·7-s − 0.900·8-s + 1.44·10-s + 11-s − 0.486·13-s − 1.59·14-s − 4.72·16-s + 2.67·17-s − 7.96·19-s + 1.16·20-s + 1.87·22-s + 7.17·23-s − 4.40·25-s − 0.912·26-s − 1.29·28-s − 4.07·29-s − 7.87·31-s − 7.07·32-s + 5.02·34-s − 0.653·35-s + 2.50·37-s − 14.9·38-s − 0.692·40-s + 1.48·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.760·4-s + 0.343·5-s − 0.321·7-s − 0.318·8-s + 0.456·10-s + 0.301·11-s − 0.134·13-s − 0.426·14-s − 1.18·16-s + 0.649·17-s − 1.82·19-s + 0.261·20-s + 0.400·22-s + 1.49·23-s − 0.881·25-s − 0.179·26-s − 0.244·28-s − 0.755·29-s − 1.41·31-s − 1.25·32-s + 0.861·34-s − 0.110·35-s + 0.411·37-s − 2.42·38-s − 0.109·40-s + 0.232·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 5 | \( 1 - 0.769T + 5T^{2} \) |
| 7 | \( 1 + 0.850T + 7T^{2} \) |
| 13 | \( 1 + 0.486T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 + 7.96T + 19T^{2} \) |
| 23 | \( 1 - 7.17T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 + 7.87T + 31T^{2} \) |
| 37 | \( 1 - 2.50T + 37T^{2} \) |
| 41 | \( 1 - 1.48T + 41T^{2} \) |
| 43 | \( 1 + 6.73T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 4.12T + 59T^{2} \) |
| 67 | \( 1 - 6.50T + 67T^{2} \) |
| 71 | \( 1 + 7.19T + 71T^{2} \) |
| 73 | \( 1 + 7.31T + 73T^{2} \) |
| 79 | \( 1 - 8.91T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 - 7.33T + 89T^{2} \) |
| 97 | \( 1 - 5.47T + 97T^{2} \) |
| 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
−7.46998256445627010890354832447, −6.73819229511604601736439631788, −6.11471232417384452515062051645, −5.54663913892689077289382440329, −4.81098186192879109332217169614, −4.05201690260762918548341997250, −3.42795281962096692809902533489, −2.58081777570751748109999340643, −1.67451635093970472217478934804, 0,
1.67451635093970472217478934804, 2.58081777570751748109999340643, 3.42795281962096692809902533489, 4.05201690260762918548341997250, 4.81098186192879109332217169614, 5.54663913892689077289382440329, 6.11471232417384452515062051645, 6.73819229511604601736439631788, 7.46998256445627010890354832447
