| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s + 4.47·7-s − 3·8-s + 9-s + 3.23·11-s − 12-s + 3.38·13-s + 4.47·14-s − 16-s − 2.85·17-s + 18-s − 3.23·19-s + 4.47·21-s + 3.23·22-s − 4.47·23-s − 3·24-s + 3.38·26-s + 27-s − 4.47·28-s + 4.38·29-s + 7.23·31-s + 5·32-s + 3.23·33-s − 2.85·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.5·4-s + 0.408·6-s + 1.69·7-s − 1.06·8-s + 0.333·9-s + 0.975·11-s − 0.288·12-s + 0.937·13-s + 1.19·14-s − 0.250·16-s − 0.692·17-s + 0.235·18-s − 0.742·19-s + 0.975·21-s + 0.689·22-s − 0.932·23-s − 0.612·24-s + 0.663·26-s + 0.192·27-s − 0.845·28-s + 0.813·29-s + 1.29·31-s + 0.883·32-s + 0.563·33-s − 0.489·34-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
| \(L(1)\) |
\(\approx\) |
\(3.282391462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.282391462\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + 2T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 8.09T + 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 5.23T + 47T^{2} \) |
| 53 | \( 1 - 1.38T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 0.618T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 - 3.09T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 3.52T + 83T^{2} \) |
| 89 | \( 1 - 7.61T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−9.002323657462668260045432006373, −8.342916251547541267608811467253, −8.062342299095762193361618769794, −6.63962792386534838733971776413, −6.00576252668532945349096965573, −4.78481468581057925511933073923, −4.38901518237591598027508741045, −3.62018348495749946622227172830, −2.32725131854991977661984969121, −1.21606054122932682573547065653,
1.21606054122932682573547065653, 2.32725131854991977661984969121, 3.62018348495749946622227172830, 4.38901518237591598027508741045, 4.78481468581057925511933073923, 6.00576252668532945349096965573, 6.63962792386534838733971776413, 8.062342299095762193361618769794, 8.342916251547541267608811467253, 9.002323657462668260045432006373
