| L(s) = 1 | + 2.05·5-s + 7-s − 5.05·11-s − 13-s + 0.273·17-s − 5.38·19-s − 5.32·23-s − 0.780·25-s − 8.32·29-s − 10.1·31-s + 2.05·35-s + 8.16·37-s − 5.05·41-s + 4.60·43-s + 1.38·47-s + 49-s + 3.43·53-s − 10.3·55-s + 1.78·59-s + 0.780·61-s − 2.05·65-s − 8.38·67-s − 7.78·71-s + 9.38·73-s − 5.05·77-s + 12.9·79-s − 5.72·83-s + ⋯ |
| L(s) = 1 | + 0.918·5-s + 0.377·7-s − 1.52·11-s − 0.277·13-s + 0.0662·17-s − 1.23·19-s − 1.11·23-s − 0.156·25-s − 1.54·29-s − 1.82·31-s + 0.347·35-s + 1.34·37-s − 0.789·41-s + 0.701·43-s + 0.201·47-s + 0.142·49-s + 0.471·53-s − 1.39·55-s + 0.231·59-s + 0.0999·61-s − 0.254·65-s − 1.02·67-s − 0.923·71-s + 1.09·73-s − 0.575·77-s + 1.45·79-s − 0.628·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 2.05T + 5T^{2} \) |
| 11 | \( 1 + 5.05T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 0.273T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + 5.32T + 23T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 8.16T + 37T^{2} \) |
| 41 | \( 1 + 5.05T + 41T^{2} \) |
| 43 | \( 1 - 4.60T + 43T^{2} \) |
| 47 | \( 1 - 1.38T + 47T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 - 1.78T + 59T^{2} \) |
| 61 | \( 1 - 0.780T + 61T^{2} \) |
| 67 | \( 1 + 8.38T + 67T^{2} \) |
| 71 | \( 1 + 7.78T + 71T^{2} \) |
| 73 | \( 1 - 9.38T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 5.72T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 2.21T + 97T^{2} \) |
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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
−8.638439904680943038917496580696, −7.81354424752232661860859631397, −7.25508347998886015476780574523, −5.97894409523226147881795956894, −5.65180050454468047630817366495, −4.73231420795417638502275338550, −3.73143099545609164630191634177, −2.38572368470452733821715638595, −1.91049359138658389164395168745, 0,
1.91049359138658389164395168745, 2.38572368470452733821715638595, 3.73143099545609164630191634177, 4.73231420795417638502275338550, 5.65180050454468047630817366495, 5.97894409523226147881795956894, 7.25508347998886015476780574523, 7.81354424752232661860859631397, 8.638439904680943038917496580696
