| L(s) = 1 | − 3·3-s − 5·5-s + 20·7-s + 9·9-s − 56·11-s − 86·13-s + 15·15-s − 106·17-s + 4·19-s − 60·21-s + 136·23-s + 25·25-s − 27·27-s − 206·29-s − 152·31-s + 168·33-s − 100·35-s + 282·37-s + 258·39-s − 246·41-s + 412·43-s − 45·45-s + 40·47-s + 57·49-s + 318·51-s − 126·53-s + 280·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.07·7-s + 1/3·9-s − 1.53·11-s − 1.83·13-s + 0.258·15-s − 1.51·17-s + 0.0482·19-s − 0.623·21-s + 1.23·23-s + 1/5·25-s − 0.192·27-s − 1.31·29-s − 0.880·31-s + 0.886·33-s − 0.482·35-s + 1.25·37-s + 1.05·39-s − 0.937·41-s + 1.46·43-s − 0.149·45-s + 0.124·47-s + 0.166·49-s + 0.873·51-s − 0.326·53-s + 0.686·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 56 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 106 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 136 T + p^{3} T^{2} \) |
| 29 | \( 1 + 206 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 282 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 40 T + p^{3} T^{2} \) |
| 53 | \( 1 + 126 T + p^{3} T^{2} \) |
| 59 | \( 1 - 56 T + p^{3} T^{2} \) |
| 61 | \( 1 + 2 T + p^{3} T^{2} \) |
| 67 | \( 1 + 388 T + p^{3} T^{2} \) |
| 71 | \( 1 + 672 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1170 T + p^{3} T^{2} \) |
| 79 | \( 1 - 408 T + p^{3} T^{2} \) |
| 83 | \( 1 - 668 T + p^{3} T^{2} \) |
| 89 | \( 1 - 66 T + p^{3} T^{2} \) |
| 97 | \( 1 + 926 T + p^{3} T^{2} \) |
| 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
−12.46463514703776210037668278964, −11.26572264432476285563810637861, −10.74754642118653737554343963329, −9.329803339228465296065782693661, −7.88380481382571055360508795733, −7.16023597148262679847623851296, −5.30655493261829368208922248556, −4.59991571390655140899607859969, −2.36344668504755659920796853295, 0,
2.36344668504755659920796853295, 4.59991571390655140899607859969, 5.30655493261829368208922248556, 7.16023597148262679847623851296, 7.88380481382571055360508795733, 9.329803339228465296065782693661, 10.74754642118653737554343963329, 11.26572264432476285563810637861, 12.46463514703776210037668278964
