| L(s) = 1 | − 2-s + 2.40·3-s + 4-s + 5-s − 2.40·6-s − 1.75·7-s − 8-s + 2.78·9-s − 10-s − 5.60·11-s + 2.40·12-s − 0.293·13-s + 1.75·14-s + 2.40·15-s + 16-s − 4.41·17-s − 2.78·18-s + 5.95·19-s + 20-s − 4.21·21-s + 5.60·22-s + 0.532·23-s − 2.40·24-s + 25-s + 0.293·26-s − 0.510·27-s − 1.75·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.38·3-s + 0.5·4-s + 0.447·5-s − 0.982·6-s − 0.662·7-s − 0.353·8-s + 0.929·9-s − 0.316·10-s − 1.69·11-s + 0.694·12-s − 0.0813·13-s + 0.468·14-s + 0.621·15-s + 0.250·16-s − 1.07·17-s − 0.657·18-s + 1.36·19-s + 0.223·20-s − 0.919·21-s + 1.19·22-s + 0.110·23-s − 0.491·24-s + 0.200·25-s + 0.0575·26-s − 0.0982·27-s − 0.331·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 7 | \( 1 + 1.75T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 + 0.293T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 - 0.532T + 23T^{2} \) |
| 29 | \( 1 - 3.48T + 29T^{2} \) |
| 31 | \( 1 + 8.31T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 3.49T + 41T^{2} \) |
| 43 | \( 1 - 9.84T + 43T^{2} \) |
| 47 | \( 1 - 6.31T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 7.19T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 2.23T + 79T^{2} \) |
| 83 | \( 1 + 4.18T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.211678961337335656409309938478, −7.38070659388281109361867539431, −7.11865914691713341996201392141, −5.87861973409694904825204872388, −5.22057354457780572662650871267, −3.98137484509200225922348131225, −2.87668056435959380789085571934, −2.69341027358928163045947533213, −1.63766370577317880636364761357, 0,
1.63766370577317880636364761357, 2.69341027358928163045947533213, 2.87668056435959380789085571934, 3.98137484509200225922348131225, 5.22057354457780572662650871267, 5.87861973409694904825204872388, 7.11865914691713341996201392141, 7.38070659388281109361867539431, 8.211678961337335656409309938478
