| L(s) = 1 | + 3-s + 3.97·5-s − 3.20·7-s + 9-s + 4.36·11-s + 5.13·13-s + 3.97·15-s + 3.01·17-s + 4.77·19-s − 3.20·21-s − 2.24·23-s + 10.7·25-s + 27-s + 7.06·29-s − 6.35·31-s + 4.36·33-s − 12.7·35-s − 0.728·37-s + 5.13·39-s + 7.71·41-s − 11.7·43-s + 3.97·45-s + 3.36·47-s + 3.27·49-s + 3.01·51-s − 8.50·53-s + 17.3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.77·5-s − 1.21·7-s + 0.333·9-s + 1.31·11-s + 1.42·13-s + 1.02·15-s + 0.730·17-s + 1.09·19-s − 0.699·21-s − 0.468·23-s + 2.15·25-s + 0.192·27-s + 1.31·29-s − 1.14·31-s + 0.759·33-s − 2.15·35-s − 0.119·37-s + 0.822·39-s + 1.20·41-s − 1.79·43-s + 0.592·45-s + 0.490·47-s + 0.467·49-s + 0.421·51-s − 1.16·53-s + 2.33·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.018863701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.018863701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 - 3.97T + 5T^{2} \) |
| 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 - 3.01T + 17T^{2} \) |
| 19 | \( 1 - 4.77T + 19T^{2} \) |
| 23 | \( 1 + 2.24T + 23T^{2} \) |
| 29 | \( 1 - 7.06T + 29T^{2} \) |
| 31 | \( 1 + 6.35T + 31T^{2} \) |
| 37 | \( 1 + 0.728T + 37T^{2} \) |
| 41 | \( 1 - 7.71T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.36T + 47T^{2} \) |
| 53 | \( 1 + 8.50T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 5.44T + 61T^{2} \) |
| 67 | \( 1 - 0.536T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 0.857T + 73T^{2} \) |
| 79 | \( 1 + 0.248T + 79T^{2} \) |
| 83 | \( 1 + 8.79T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−8.289331420881883430297099733578, −7.15205671046747466636490372179, −6.51496083427474285202183277683, −6.04246271693594327077860701362, −5.50869150512336357367144081341, −4.31325607520125242160310905313, −3.33650805488266128141063964795, −2.98079067299160391630442329061, −1.69398883697004106963008075950, −1.17850732101327362903905960156,
1.17850732101327362903905960156, 1.69398883697004106963008075950, 2.98079067299160391630442329061, 3.33650805488266128141063964795, 4.31325607520125242160310905313, 5.50869150512336357367144081341, 6.04246271693594327077860701362, 6.51496083427474285202183277683, 7.15205671046747466636490372179, 8.289331420881883430297099733578
