| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2.73·7-s + 8-s + 9-s + 5.19·11-s − 12-s + 4.73·13-s + 2.73·14-s + 16-s − 2.73·17-s + 18-s + 19-s − 2.73·21-s + 5.19·22-s + 8.46·23-s − 24-s + 4.73·26-s − 27-s + 2.73·28-s − 9.19·29-s − 5.92·31-s + 32-s − 5.19·33-s − 2.73·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.03·7-s + 0.353·8-s + 0.333·9-s + 1.56·11-s − 0.288·12-s + 1.31·13-s + 0.730·14-s + 0.250·16-s − 0.662·17-s + 0.235·18-s + 0.229·19-s − 0.596·21-s + 1.10·22-s + 1.76·23-s − 0.204·24-s + 0.928·26-s − 0.192·27-s + 0.516·28-s − 1.70·29-s − 1.06·31-s + 0.176·32-s − 0.904·33-s − 0.468·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.252620128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.252620128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 23 | \( 1 - 8.46T + 23T^{2} \) |
| 29 | \( 1 + 9.19T + 29T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 - 8.26T + 61T^{2} \) |
| 67 | \( 1 + 5.19T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.983393638329352776948390823111, −7.88259923770659831535443333040, −6.99688805945246289905831484142, −6.48653844734527045112358021653, −5.57697346970628539897524518216, −4.97491738302514296442542307338, −4.02340333180061077855155384203, −3.49628878613053952275137333029, −1.88897408435967492974056121666, −1.17707155491389899811605176996,
1.17707155491389899811605176996, 1.88897408435967492974056121666, 3.49628878613053952275137333029, 4.02340333180061077855155384203, 4.97491738302514296442542307338, 5.57697346970628539897524518216, 6.48653844734527045112358021653, 6.99688805945246289905831484142, 7.88259923770659831535443333040, 8.983393638329352776948390823111
