| L(s) = 1 | − 5·5-s + 30·7-s − 50·11-s − 20·13-s − 10·17-s + 44·19-s − 120·23-s + 25·25-s − 50·29-s − 108·31-s − 150·35-s − 40·37-s + 400·41-s − 280·43-s + 280·47-s + 557·49-s − 610·53-s + 250·55-s − 50·59-s − 518·61-s + 100·65-s + 180·67-s − 700·71-s − 410·73-s − 1.50e3·77-s + 516·79-s − 660·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.61·7-s − 1.37·11-s − 0.426·13-s − 0.142·17-s + 0.531·19-s − 1.08·23-s + 1/5·25-s − 0.320·29-s − 0.625·31-s − 0.724·35-s − 0.177·37-s + 1.52·41-s − 0.993·43-s + 0.868·47-s + 1.62·49-s − 1.58·53-s + 0.612·55-s − 0.110·59-s − 1.08·61-s + 0.190·65-s + 0.328·67-s − 1.17·71-s − 0.657·73-s − 2.22·77-s + 0.734·79-s − 0.872·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 30 T + p^{3} T^{2} \) |
| 11 | \( 1 + 50 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 10 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 108 T + p^{3} T^{2} \) |
| 37 | \( 1 + 40 T + p^{3} T^{2} \) |
| 41 | \( 1 - 400 T + p^{3} T^{2} \) |
| 43 | \( 1 + 280 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 + 610 T + p^{3} T^{2} \) |
| 59 | \( 1 + 50 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 180 T + p^{3} T^{2} \) |
| 71 | \( 1 + 700 T + p^{3} T^{2} \) |
| 73 | \( 1 + 410 T + p^{3} T^{2} \) |
| 79 | \( 1 - 516 T + p^{3} T^{2} \) |
| 83 | \( 1 + 660 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1500 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1630 T + p^{3} T^{2} \) |
| 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
−9.625066486433990990711406786295, −8.474981396335358201336889628879, −7.82284654051743822351947759237, −7.33407127100448864218027965786, −5.76778910503619452063112596813, −4.99947286230949706833474886582, −4.18655386921859148875250455373, −2.73505116495986139671841835703, −1.61122645229544698110237576747, 0,
1.61122645229544698110237576747, 2.73505116495986139671841835703, 4.18655386921859148875250455373, 4.99947286230949706833474886582, 5.76778910503619452063112596813, 7.33407127100448864218027965786, 7.82284654051743822351947759237, 8.474981396335358201336889628879, 9.625066486433990990711406786295
