| L(s) = 1 | + 4·5-s − 4·7-s − 26·11-s + 70·13-s − 94·17-s + 54·19-s + 23·23-s − 109·25-s + 86·29-s − 144·31-s − 16·35-s − 172·37-s + 42·41-s + 386·43-s + 80·47-s − 327·49-s + 108·53-s − 104·55-s − 164·59-s − 400·61-s + 280·65-s + 398·67-s + 320·71-s − 810·73-s + 104·77-s − 204·79-s − 102·83-s + ⋯ |
| L(s) = 1 | + 0.357·5-s − 0.215·7-s − 0.712·11-s + 1.49·13-s − 1.34·17-s + 0.652·19-s + 0.208·23-s − 0.871·25-s + 0.550·29-s − 0.834·31-s − 0.0772·35-s − 0.764·37-s + 0.159·41-s + 1.36·43-s + 0.248·47-s − 0.953·49-s + 0.279·53-s − 0.254·55-s − 0.361·59-s − 0.839·61-s + 0.534·65-s + 0.725·67-s + 0.534·71-s − 1.29·73-s + 0.153·77-s − 0.290·79-s − 0.134·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - p T \) |
| good | 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + 94 T + p^{3} T^{2} \) |
| 19 | \( 1 - 54 T + p^{3} T^{2} \) |
| 29 | \( 1 - 86 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 + 172 T + p^{3} T^{2} \) |
| 41 | \( 1 - 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 386 T + p^{3} T^{2} \) |
| 47 | \( 1 - 80 T + p^{3} T^{2} \) |
| 53 | \( 1 - 108 T + p^{3} T^{2} \) |
| 59 | \( 1 + 164 T + p^{3} T^{2} \) |
| 61 | \( 1 + 400 T + p^{3} T^{2} \) |
| 67 | \( 1 - 398 T + p^{3} T^{2} \) |
| 71 | \( 1 - 320 T + p^{3} T^{2} \) |
| 73 | \( 1 + 810 T + p^{3} T^{2} \) |
| 79 | \( 1 + 204 T + p^{3} T^{2} \) |
| 83 | \( 1 + 102 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1018 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1370 T + p^{3} T^{2} \) |
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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
−8.709749704732410255171115206395, −7.86242739099988897363235015786, −6.95244122723034266819489521432, −6.12094793690819963591676697335, −5.46884660983174675770255298588, −4.39374671637082497673610817385, −3.47175722530017832192190367850, −2.43505697356376981913417518015, −1.35394409422223165545378255190, 0,
1.35394409422223165545378255190, 2.43505697356376981913417518015, 3.47175722530017832192190367850, 4.39374671637082497673610817385, 5.46884660983174675770255298588, 6.12094793690819963591676697335, 6.95244122723034266819489521432, 7.86242739099988897363235015786, 8.709749704732410255171115206395
