| L(s) = 1 | − 1.12·2-s + 3-s − 0.740·4-s − 1.12·6-s + 1.11·7-s + 3.07·8-s + 9-s + 3.67·11-s − 0.740·12-s − 4.05·13-s − 1.24·14-s − 1.97·16-s + 2.12·17-s − 1.12·18-s − 4.06·19-s + 1.11·21-s − 4.11·22-s + 6.17·23-s + 3.07·24-s + 4.54·26-s + 27-s − 0.824·28-s − 2.25·29-s + 10.0·31-s − 3.93·32-s + 3.67·33-s − 2.38·34-s + ⋯ |
| L(s) = 1 | − 0.793·2-s + 0.577·3-s − 0.370·4-s − 0.458·6-s + 0.420·7-s + 1.08·8-s + 0.333·9-s + 1.10·11-s − 0.213·12-s − 1.12·13-s − 0.334·14-s − 0.492·16-s + 0.514·17-s − 0.264·18-s − 0.931·19-s + 0.243·21-s − 0.878·22-s + 1.28·23-s + 0.627·24-s + 0.891·26-s + 0.192·27-s − 0.155·28-s − 0.419·29-s + 1.80·31-s − 0.696·32-s + 0.638·33-s − 0.408·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359553010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359553010\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.12T + 2T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 - 2.12T + 17T^{2} \) |
| 19 | \( 1 + 4.06T + 19T^{2} \) |
| 23 | \( 1 - 6.17T + 23T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 + 3.50T + 53T^{2} \) |
| 59 | \( 1 + 6.59T + 59T^{2} \) |
| 61 | \( 1 + 9.10T + 61T^{2} \) |
| 67 | \( 1 - 2.62T + 67T^{2} \) |
| 71 | \( 1 - 0.660T + 71T^{2} \) |
| 73 | \( 1 - 7.47T + 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−9.317496466483100998466286388742, −8.480472240547853023398991941350, −7.890133502046111237129512442470, −7.15547093356250528323766880126, −6.26097941179739414795313788090, −4.84035549041700819034464343803, −4.44071315638205817433866103979, −3.24397678861649761029938054500, −2.01194103234678130750228730253, −0.913290204547767087296950666723,
0.913290204547767087296950666723, 2.01194103234678130750228730253, 3.24397678861649761029938054500, 4.44071315638205817433866103979, 4.84035549041700819034464343803, 6.26097941179739414795313788090, 7.15547093356250528323766880126, 7.890133502046111237129512442470, 8.480472240547853023398991941350, 9.317496466483100998466286388742
