| L(s) = 1 | + 3·3-s − 5·5-s + 12·7-s + 9·9-s − 20·11-s − 58·13-s − 15·15-s − 70·17-s − 92·19-s + 36·21-s + 112·23-s + 25·25-s + 27·27-s + 66·29-s − 108·31-s − 60·33-s − 60·35-s − 58·37-s − 174·39-s + 66·41-s − 388·43-s − 45·45-s − 408·47-s − 199·49-s − 210·51-s + 474·53-s + 100·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.647·7-s + 1/3·9-s − 0.548·11-s − 1.23·13-s − 0.258·15-s − 0.998·17-s − 1.11·19-s + 0.374·21-s + 1.01·23-s + 1/5·25-s + 0.192·27-s + 0.422·29-s − 0.625·31-s − 0.316·33-s − 0.289·35-s − 0.257·37-s − 0.714·39-s + 0.251·41-s − 1.37·43-s − 0.149·45-s − 1.26·47-s − 0.580·49-s − 0.576·51-s + 1.22·53-s + 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 112 T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 + 108 T + p^{3} T^{2} \) |
| 37 | \( 1 + 58 T + p^{3} T^{2} \) |
| 41 | \( 1 - 66 T + p^{3} T^{2} \) |
| 43 | \( 1 + 388 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 - 474 T + p^{3} T^{2} \) |
| 59 | \( 1 + 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 14 T + p^{3} T^{2} \) |
| 67 | \( 1 + 276 T + p^{3} T^{2} \) |
| 71 | \( 1 + 96 T + p^{3} T^{2} \) |
| 73 | \( 1 + 790 T + p^{3} T^{2} \) |
| 79 | \( 1 - 308 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1210 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1426 T + p^{3} T^{2} \) |
| 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
−10.18657247237315025526598318746, −9.064734531183630197415601939942, −8.338073699289670873899991242992, −7.49450919565752592596768267580, −6.65365261340905937795637624838, −5.06694045913535504274332925898, −4.39352276759411141549194308534, −2.97927319353961298355927674433, −1.90161079809174123446594632977, 0,
1.90161079809174123446594632977, 2.97927319353961298355927674433, 4.39352276759411141549194308534, 5.06694045913535504274332925898, 6.65365261340905937795637624838, 7.49450919565752592596768267580, 8.338073699289670873899991242992, 9.064734531183630197415601939942, 10.18657247237315025526598318746
