| L(s) = 1 | − 3-s + 0.618·5-s − 7-s + 9-s + 4.23·11-s − 0.381·13-s − 0.618·15-s − 4.23·17-s − 0.236·19-s + 21-s − 23-s − 4.61·25-s − 27-s − 0.236·29-s + 8.23·31-s − 4.23·33-s − 0.618·35-s − 8.70·37-s + 0.381·39-s + 4.70·41-s + 4.09·43-s + 0.618·45-s + 0.763·47-s + 49-s + 4.23·51-s − 4.61·53-s + 2.61·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.276·5-s − 0.377·7-s + 0.333·9-s + 1.27·11-s − 0.105·13-s − 0.159·15-s − 1.02·17-s − 0.0541·19-s + 0.218·21-s − 0.208·23-s − 0.923·25-s − 0.192·27-s − 0.0438·29-s + 1.47·31-s − 0.737·33-s − 0.104·35-s − 1.43·37-s + 0.0611·39-s + 0.735·41-s + 0.623·43-s + 0.0921·45-s + 0.111·47-s + 0.142·49-s + 0.593·51-s − 0.634·53-s + 0.353·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.618T + 5T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 0.381T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 + 0.236T + 19T^{2} \) |
| 29 | \( 1 + 0.236T + 29T^{2} \) |
| 31 | \( 1 - 8.23T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 - 4.70T + 41T^{2} \) |
| 43 | \( 1 - 4.09T + 43T^{2} \) |
| 47 | \( 1 - 0.763T + 47T^{2} \) |
| 53 | \( 1 + 4.61T + 53T^{2} \) |
| 59 | \( 1 + 7.56T + 59T^{2} \) |
| 61 | \( 1 + 3.14T + 61T^{2} \) |
| 67 | \( 1 - 0.145T + 67T^{2} \) |
| 71 | \( 1 + 5.61T + 71T^{2} \) |
| 73 | \( 1 + 7.47T + 73T^{2} \) |
| 79 | \( 1 - 7T + 79T^{2} \) |
| 83 | \( 1 + 5.94T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 - 5.94T + 97T^{2} \) |
| 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
−7.34630239274250066569856885926, −6.64716112913360909156894641723, −6.22592715125066693847856357878, −5.58163562303159966956209160662, −4.54742457145212628554808723583, −4.12627422298739473447635188753, −3.15584743339247270670052329856, −2.12101817543015349362584851419, −1.24319181032224879240721055125, 0,
1.24319181032224879240721055125, 2.12101817543015349362584851419, 3.15584743339247270670052329856, 4.12627422298739473447635188753, 4.54742457145212628554808723583, 5.58163562303159966956209160662, 6.22592715125066693847856357878, 6.64716112913360909156894641723, 7.34630239274250066569856885926
