| L(s) = 1 | + 3-s − 2·4-s + 3·5-s − 7-s + 9-s − 6·11-s − 2·12-s − 2·13-s + 3·15-s + 4·16-s − 2·17-s + 6·19-s − 6·20-s − 21-s − 8·23-s + 4·25-s + 27-s + 2·28-s + 8·29-s + 2·31-s − 6·33-s − 3·35-s − 2·36-s − 7·37-s − 2·39-s − 7·41-s + 12·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.577·12-s − 0.554·13-s + 0.774·15-s + 16-s − 0.485·17-s + 1.37·19-s − 1.34·20-s − 0.218·21-s − 1.66·23-s + 4/5·25-s + 0.192·27-s + 0.377·28-s + 1.48·29-s + 0.359·31-s − 1.04·33-s − 0.507·35-s − 1/3·36-s − 1.15·37-s − 0.320·39-s − 1.09·41-s + 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38829 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38829 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.666718129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666718129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.57020002240121, −14.01151653957593, −13.73696451330398, −13.53860831069759, −12.92862132211174, −12.30767987590629, −12.03526969235350, −10.83959486753518, −10.20257050155308, −10.09802107563828, −9.581601510177293, −9.119958137189449, −8.307166747972940, −8.084324094954407, −7.378002120474847, −6.656426016562176, −5.929463531224753, −5.346597049779134, −5.040174390567420, −4.333716187923688, −3.459869553055468, −2.812390206119584, −2.323170758798519, −1.528684180892295, −0.4494612483778204,
0.4494612483778204, 1.528684180892295, 2.323170758798519, 2.812390206119584, 3.459869553055468, 4.333716187923688, 5.040174390567420, 5.346597049779134, 5.929463531224753, 6.656426016562176, 7.378002120474847, 8.084324094954407, 8.307166747972940, 9.119958137189449, 9.581601510177293, 10.09802107563828, 10.20257050155308, 10.83959486753518, 12.03526969235350, 12.30767987590629, 12.92862132211174, 13.53860831069759, 13.73696451330398, 14.01151653957593, 14.57020002240121
