| L(s) = 1 | + 2·7-s − 39·11-s + 84·13-s + 61·17-s + 151·19-s + 58·23-s − 192·29-s − 18·31-s − 138·37-s − 229·41-s − 164·43-s + 212·47-s − 339·49-s − 578·53-s + 336·59-s + 858·61-s − 209·67-s + 780·71-s − 403·73-s − 78·77-s − 230·79-s + 1.29e3·83-s + 1.36e3·89-s + 168·91-s + 382·97-s + 794·101-s − 1.34e3·103-s + ⋯ |
| L(s) = 1 | + 0.107·7-s − 1.06·11-s + 1.79·13-s + 0.870·17-s + 1.82·19-s + 0.525·23-s − 1.22·29-s − 0.104·31-s − 0.613·37-s − 0.872·41-s − 0.581·43-s + 0.657·47-s − 0.988·49-s − 1.49·53-s + 0.741·59-s + 1.80·61-s − 0.381·67-s + 1.30·71-s − 0.646·73-s − 0.115·77-s − 0.327·79-s + 1.70·83-s + 1.63·89-s + 0.193·91-s + 0.399·97-s + 0.782·101-s − 1.28·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.425997890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.425997890\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 39 T + p^{3} T^{2} \) |
| 13 | \( 1 - 84 T + p^{3} T^{2} \) |
| 17 | \( 1 - 61 T + p^{3} T^{2} \) |
| 19 | \( 1 - 151 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 192 T + p^{3} T^{2} \) |
| 31 | \( 1 + 18 T + p^{3} T^{2} \) |
| 37 | \( 1 + 138 T + p^{3} T^{2} \) |
| 41 | \( 1 + 229 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 212 T + p^{3} T^{2} \) |
| 53 | \( 1 + 578 T + p^{3} T^{2} \) |
| 59 | \( 1 - 336 T + p^{3} T^{2} \) |
| 61 | \( 1 - 858 T + p^{3} T^{2} \) |
| 67 | \( 1 + 209 T + p^{3} T^{2} \) |
| 71 | \( 1 - 780 T + p^{3} T^{2} \) |
| 73 | \( 1 + 403 T + p^{3} T^{2} \) |
| 79 | \( 1 + 230 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1293 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1369 T + p^{3} T^{2} \) |
| 97 | \( 1 - 382 T + p^{3} T^{2} \) |
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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
−8.878889426657419013543641062115, −8.052690011827743019192469753472, −7.50032109289679748841680785364, −6.50256696396724692577270216045, −5.51204535605905975126701562866, −5.10698663899271908694077123044, −3.63788041321660002683775181232, −3.17939330700458207203193074587, −1.74732386624454406107512523403, −0.76476213121997204688517796851,
0.76476213121997204688517796851, 1.74732386624454406107512523403, 3.17939330700458207203193074587, 3.63788041321660002683775181232, 5.10698663899271908694077123044, 5.51204535605905975126701562866, 6.50256696396724692577270216045, 7.50032109289679748841680785364, 8.052690011827743019192469753472, 8.878889426657419013543641062115
