| L(s) = 1 | − 1.04·5-s + 0.449·7-s + 1.28·11-s + 2·13-s + 2.09·17-s − 19-s − 3.61·23-s − 3.89·25-s − 3.38·29-s − 2.55·31-s − 0.471·35-s − 0.449·37-s + 11.6·41-s − 9.34·43-s − 2.33·47-s − 6.79·49-s + 11.2·53-s − 1.34·55-s − 1.52·59-s + 6.34·61-s − 2.09·65-s − 12.3·67-s − 13.5·71-s + 6.34·73-s + 0.577·77-s − 5·79-s + 4.43·83-s + ⋯ |
| L(s) = 1 | − 0.469·5-s + 0.169·7-s + 0.387·11-s + 0.554·13-s + 0.508·17-s − 0.229·19-s − 0.754·23-s − 0.779·25-s − 0.628·29-s − 0.458·31-s − 0.0797·35-s − 0.0738·37-s + 1.82·41-s − 1.42·43-s − 0.340·47-s − 0.971·49-s + 1.53·53-s − 0.181·55-s − 0.198·59-s + 0.812·61-s − 0.260·65-s − 1.50·67-s − 1.60·71-s + 0.743·73-s + 0.0658·77-s − 0.562·79-s + 0.486·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 1.04T + 5T^{2} \) |
| 7 | \( 1 - 0.449T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 + 3.38T + 29T^{2} \) |
| 31 | \( 1 + 2.55T + 31T^{2} \) |
| 37 | \( 1 + 0.449T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 9.34T + 43T^{2} \) |
| 47 | \( 1 + 2.33T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 6.34T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 6.34T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 4.43T + 83T^{2} \) |
| 89 | \( 1 + 5.95T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
| 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
−7.56773199213531091374305591908, −6.83152247420254172125848682020, −6.00492767341049937342780820707, −5.52578065174730684083984905290, −4.48716402351884805994596084288, −3.90777289813260141441467851141, −3.25260955028930628227212417631, −2.14624479007438700459904484442, −1.27694959651934920351804875411, 0,
1.27694959651934920351804875411, 2.14624479007438700459904484442, 3.25260955028930628227212417631, 3.90777289813260141441467851141, 4.48716402351884805994596084288, 5.52578065174730684083984905290, 6.00492767341049937342780820707, 6.83152247420254172125848682020, 7.56773199213531091374305591908
