| L(s) = 1 | − 2.70·2-s − 3-s + 5.34·4-s + 2.70·6-s − 9.04·8-s + 9-s + 2·11-s − 5.34·12-s − 0.921·13-s + 13.8·16-s − 1.07·17-s − 2.70·18-s − 3.07·19-s − 5.41·22-s + 2.34·23-s + 9.04·24-s + 2.49·26-s − 27-s − 6.68·29-s + 7.75·31-s − 19.3·32-s − 2·33-s + 2.92·34-s + 5.34·36-s + 10.8·37-s + 8.34·38-s + 0.921·39-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 0.577·3-s + 2.67·4-s + 1.10·6-s − 3.19·8-s + 0.333·9-s + 0.603·11-s − 1.54·12-s − 0.255·13-s + 3.45·16-s − 0.261·17-s − 0.638·18-s − 0.706·19-s − 1.15·22-s + 0.487·23-s + 1.84·24-s + 0.489·26-s − 0.192·27-s − 1.24·29-s + 1.39·31-s − 3.42·32-s − 0.348·33-s + 0.501·34-s + 0.890·36-s + 1.78·37-s + 1.35·38-s + 0.147·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 0.921T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 6.49T + 41T^{2} \) |
| 43 | \( 1 + 6.52T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 4.15T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 + 6.83T + 83T^{2} \) |
| 89 | \( 1 + 8.34T + 89T^{2} \) |
| 97 | \( 1 - 8.43T + 97T^{2} \) |
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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
−8.218279476079218068805357583810, −7.62624284955739826504199252108, −6.67477566965535372968847849189, −6.47058772692979905940239932019, −5.46864384269813229538195131328, −4.29642790110606481260960268817, −3.04961765692660278457060654845, −2.04268395897040096406739916150, −1.13380744699443248209789838169, 0,
1.13380744699443248209789838169, 2.04268395897040096406739916150, 3.04961765692660278457060654845, 4.29642790110606481260960268817, 5.46864384269813229538195131328, 6.47058772692979905940239932019, 6.67477566965535372968847849189, 7.62624284955739826504199252108, 8.218279476079218068805357583810
