| L(s) = 1 | + 2.64·5-s + 4.37·7-s − 3.58·11-s − 6.58·13-s + 1.73·17-s − 2.55·19-s + 7.58·23-s + 2.00·25-s + 6.10·29-s + 8.75·31-s + 11.5·35-s + 2.58·37-s − 1.82·41-s + 2.55·43-s + 8·47-s + 12.1·49-s + 1.82·53-s − 9.47·55-s + 8·59-s − 1.41·61-s − 17.4·65-s + 2.55·67-s + 0.417·71-s − 6.16·73-s − 15.6·77-s + 9.47·79-s + 15.1·83-s + ⋯ |
| L(s) = 1 | + 1.18·5-s + 1.65·7-s − 1.08·11-s − 1.82·13-s + 0.420·17-s − 0.585·19-s + 1.58·23-s + 0.400·25-s + 1.13·29-s + 1.57·31-s + 1.95·35-s + 0.424·37-s − 0.285·41-s + 0.388·43-s + 1.16·47-s + 1.73·49-s + 0.251·53-s − 1.27·55-s + 1.04·59-s − 0.181·61-s − 2.16·65-s + 0.311·67-s + 0.0495·71-s − 0.721·73-s − 1.78·77-s + 1.06·79-s + 1.66·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.576249575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.576249575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.64T + 5T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 + 6.58T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 2.55T + 19T^{2} \) |
| 23 | \( 1 - 7.58T + 23T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 31 | \( 1 - 8.75T + 31T^{2} \) |
| 37 | \( 1 - 2.58T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 - 2.55T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 1.82T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 - 2.55T + 67T^{2} \) |
| 71 | \( 1 - 0.417T + 71T^{2} \) |
| 73 | \( 1 + 6.16T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 5.16T + 97T^{2} \) |
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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
−8.839108134385151972460222952589, −8.072155594231840799870282326906, −7.48163028221718010838295218834, −6.61368851286835753435581884405, −5.48753397233011580630329976204, −5.03935734986125837504910557106, −4.49279648245163491476590927808, −2.64945040223546785627414318880, −2.31901180165068005305668461465, −1.06559269239501104444819619026,
1.06559269239501104444819619026, 2.31901180165068005305668461465, 2.64945040223546785627414318880, 4.49279648245163491476590927808, 5.03935734986125837504910557106, 5.48753397233011580630329976204, 6.61368851286835753435581884405, 7.48163028221718010838295218834, 8.072155594231840799870282326906, 8.839108134385151972460222952589
