| L(s) = 1 | − 5·5-s + 24·7-s + 52·11-s + 22·13-s + 14·17-s + 20·19-s − 168·23-s + 25·25-s − 230·29-s + 288·31-s − 120·35-s − 34·37-s − 122·41-s + 188·43-s + 256·47-s + 233·49-s + 338·53-s − 260·55-s + 100·59-s + 742·61-s − 110·65-s + 84·67-s − 328·71-s − 38·73-s + 1.24e3·77-s + 240·79-s + 1.21e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.29·7-s + 1.42·11-s + 0.469·13-s + 0.199·17-s + 0.241·19-s − 1.52·23-s + 1/5·25-s − 1.47·29-s + 1.66·31-s − 0.579·35-s − 0.151·37-s − 0.464·41-s + 0.666·43-s + 0.794·47-s + 0.679·49-s + 0.875·53-s − 0.637·55-s + 0.220·59-s + 1.55·61-s − 0.209·65-s + 0.153·67-s − 0.548·71-s − 0.0609·73-s + 1.84·77-s + 0.341·79-s + 1.60·83-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
| \(L(2)\) |
\(\approx\) |
\(2.466945721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.466945721\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 230 T + p^{3} T^{2} \) |
| 31 | \( 1 - 288 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 122 T + p^{3} T^{2} \) |
| 43 | \( 1 - 188 T + p^{3} T^{2} \) |
| 47 | \( 1 - 256 T + p^{3} T^{2} \) |
| 53 | \( 1 - 338 T + p^{3} T^{2} \) |
| 59 | \( 1 - 100 T + p^{3} T^{2} \) |
| 61 | \( 1 - 742 T + p^{3} T^{2} \) |
| 67 | \( 1 - 84 T + p^{3} T^{2} \) |
| 71 | \( 1 + 328 T + p^{3} T^{2} \) |
| 73 | \( 1 + 38 T + p^{3} T^{2} \) |
| 79 | \( 1 - 240 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1212 T + p^{3} T^{2} \) |
| 89 | \( 1 + 330 T + p^{3} T^{2} \) |
| 97 | \( 1 - 866 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.03592655644087435448539811614, −9.023850903589782693711853777871, −8.262917398673117374620972709533, −7.55547112557296945683443887016, −6.49471603265344704319544748879, −5.51254354899710529348763514242, −4.35064539347210923658634660782, −3.70989584895754066074388930148, −2.03007275694702051065720781230, −0.965873354858502213873430288091,
0.965873354858502213873430288091, 2.03007275694702051065720781230, 3.70989584895754066074388930148, 4.35064539347210923658634660782, 5.51254354899710529348763514242, 6.49471603265344704319544748879, 7.55547112557296945683443887016, 8.262917398673117374620972709533, 9.023850903589782693711853777871, 10.03592655644087435448539811614
