| L(s) = 1 | + 3-s − 5-s + 7-s − 2·9-s + 13-s − 15-s − 3·17-s + 19-s + 21-s − 3·23-s + 25-s − 5·27-s + 3·29-s − 2·31-s − 35-s + 10·37-s + 39-s + 6·41-s + 2·43-s + 2·45-s − 6·49-s − 3·51-s − 3·53-s + 57-s + 3·59-s − 8·61-s − 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.277·13-s − 0.258·15-s − 0.727·17-s + 0.229·19-s + 0.218·21-s − 0.625·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s − 0.359·31-s − 0.169·35-s + 1.64·37-s + 0.160·39-s + 0.937·41-s + 0.304·43-s + 0.298·45-s − 6/7·49-s − 0.420·51-s − 0.412·53-s + 0.132·57-s + 0.390·59-s − 1.02·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.87612912524915938223314087027, −7.19726440784475341058029871875, −6.21528861087112814565046241311, −5.69107960408859207489557272311, −4.60293887045679369072002575929, −4.10124977314440080814556939276, −3.09838209289024367163880459826, −2.49598019406012877403916200737, −1.39627742076520574651487592934, 0,
1.39627742076520574651487592934, 2.49598019406012877403916200737, 3.09838209289024367163880459826, 4.10124977314440080814556939276, 4.60293887045679369072002575929, 5.69107960408859207489557272311, 6.21528861087112814565046241311, 7.19726440784475341058029871875, 7.87612912524915938223314087027
