| L(s) = 1 | − 2-s + 3-s + 4-s − 2.57·5-s − 6-s − 4.81·7-s − 8-s + 9-s + 2.57·10-s + 1.68·11-s + 12-s − 13-s + 4.81·14-s − 2.57·15-s + 16-s + 6.18·17-s − 18-s + 0.771·19-s − 2.57·20-s − 4.81·21-s − 1.68·22-s − 2.46·23-s − 24-s + 1.61·25-s + 26-s + 27-s − 4.81·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.15·5-s − 0.408·6-s − 1.81·7-s − 0.353·8-s + 0.333·9-s + 0.813·10-s + 0.506·11-s + 0.288·12-s − 0.277·13-s + 1.28·14-s − 0.664·15-s + 0.250·16-s + 1.49·17-s − 0.235·18-s + 0.176·19-s − 0.575·20-s − 1.05·21-s − 0.358·22-s − 0.513·23-s − 0.204·24-s + 0.323·25-s + 0.196·26-s + 0.192·27-s − 0.909·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 2.57T + 5T^{2} \) |
| 7 | \( 1 + 4.81T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 17 | \( 1 - 6.18T + 17T^{2} \) |
| 19 | \( 1 - 0.771T + 19T^{2} \) |
| 23 | \( 1 + 2.46T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 - 7.10T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 4.87T + 43T^{2} \) |
| 47 | \( 1 + 7.45T + 47T^{2} \) |
| 53 | \( 1 - 4.54T + 53T^{2} \) |
| 59 | \( 1 + 5.91T + 59T^{2} \) |
| 61 | \( 1 + 0.0881T + 61T^{2} \) |
| 67 | \( 1 + 9.81T + 67T^{2} \) |
| 71 | \( 1 + 6.15T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 + 3.92T + 79T^{2} \) |
| 83 | \( 1 + 1.22T + 83T^{2} \) |
| 89 | \( 1 - 6.45T + 89T^{2} \) |
| 97 | \( 1 + 8.09T + 97T^{2} \) |
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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
−7.51518700566527926179778611806, −7.17512795077838866761794970973, −6.18790844689412588354932227330, −5.72008324008686738316394211041, −4.32143172631233108310964227148, −3.58553753459048357442804218170, −3.26482777101139502590695638719, −2.32986902366311676861853963242, −0.989210729909969253617787556756, 0,
0.989210729909969253617787556756, 2.32986902366311676861853963242, 3.26482777101139502590695638719, 3.58553753459048357442804218170, 4.32143172631233108310964227148, 5.72008324008686738316394211041, 6.18790844689412588354932227330, 7.17512795077838866761794970973, 7.51518700566527926179778611806
