| L(s) = 1 | − 2-s + 4-s − 3.79·5-s − 2.04·7-s − 8-s + 3.79·10-s + 3.58·11-s − 1.68·13-s + 2.04·14-s + 16-s − 7.06·17-s + 6.27·19-s − 3.79·20-s − 3.58·22-s − 5.91·23-s + 9.37·25-s + 1.68·26-s − 2.04·28-s − 1.71·29-s − 6.59·31-s − 32-s + 7.06·34-s + 7.75·35-s − 6.84·37-s − 6.27·38-s + 3.79·40-s − 7.17·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.69·5-s − 0.773·7-s − 0.353·8-s + 1.19·10-s + 1.08·11-s − 0.466·13-s + 0.546·14-s + 0.250·16-s − 1.71·17-s + 1.43·19-s − 0.847·20-s − 0.764·22-s − 1.23·23-s + 1.87·25-s + 0.330·26-s − 0.386·28-s − 0.317·29-s − 1.18·31-s − 0.176·32-s + 1.21·34-s + 1.31·35-s − 1.12·37-s − 1.01·38-s + 0.599·40-s − 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3778346481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3778346481\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 3.79T + 5T^{2} \) |
| 7 | \( 1 + 2.04T + 7T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 13 | \( 1 + 1.68T + 13T^{2} \) |
| 17 | \( 1 + 7.06T + 17T^{2} \) |
| 19 | \( 1 - 6.27T + 19T^{2} \) |
| 23 | \( 1 + 5.91T + 23T^{2} \) |
| 29 | \( 1 + 1.71T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 + 6.84T + 37T^{2} \) |
| 41 | \( 1 + 7.17T + 41T^{2} \) |
| 43 | \( 1 - 8.20T + 43T^{2} \) |
| 47 | \( 1 + 4.20T + 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 2.28T + 61T^{2} \) |
| 67 | \( 1 + 9.15T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 0.168T + 73T^{2} \) |
| 79 | \( 1 - 4.46T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 8.00T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.447762030824287499398842073962, −7.73854315356308584980136078260, −6.98270014490855676194643686775, −6.72074565624622370463756086423, −5.56208124310393267026620848731, −4.41682385637001202062519445281, −3.76174406293905721617945093936, −3.09981768518117415271961038573, −1.81038240964575695895625880099, −0.37957003176339872126792791663,
0.37957003176339872126792791663, 1.81038240964575695895625880099, 3.09981768518117415271961038573, 3.76174406293905721617945093936, 4.41682385637001202062519445281, 5.56208124310393267026620848731, 6.72074565624622370463756086423, 6.98270014490855676194643686775, 7.73854315356308584980136078260, 8.447762030824287499398842073962
