| L(s) = 1 | − 0.759·2-s + 2.25·3-s − 1.42·4-s − 1.71·6-s − 4.95·7-s + 2.59·8-s + 2.09·9-s − 11-s − 3.21·12-s − 0.499·13-s + 3.75·14-s + 0.874·16-s + 4.34·17-s − 1.59·18-s − 19-s − 11.1·21-s + 0.759·22-s + 2.78·23-s + 5.86·24-s + 0.378·26-s − 2.03·27-s + 7.05·28-s + 8.84·29-s + 1.67·31-s − 5.86·32-s − 2.25·33-s − 3.30·34-s + ⋯ |
| L(s) = 1 | − 0.536·2-s + 1.30·3-s − 0.711·4-s − 0.699·6-s − 1.87·7-s + 0.918·8-s + 0.699·9-s − 0.301·11-s − 0.928·12-s − 0.138·13-s + 1.00·14-s + 0.218·16-s + 1.05·17-s − 0.375·18-s − 0.229·19-s − 2.44·21-s + 0.161·22-s + 0.581·23-s + 1.19·24-s + 0.0743·26-s − 0.391·27-s + 1.33·28-s + 1.64·29-s + 0.300·31-s − 1.03·32-s − 0.393·33-s − 0.566·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.759T + 2T^{2} \) |
| 3 | \( 1 - 2.25T + 3T^{2} \) |
| 7 | \( 1 + 4.95T + 7T^{2} \) |
| 13 | \( 1 + 0.499T + 13T^{2} \) |
| 17 | \( 1 - 4.34T + 17T^{2} \) |
| 23 | \( 1 - 2.78T + 23T^{2} \) |
| 29 | \( 1 - 8.84T + 29T^{2} \) |
| 31 | \( 1 - 1.67T + 31T^{2} \) |
| 37 | \( 1 - 1.81T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 2.65T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 2.84T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 7.25T + 89T^{2} \) |
| 97 | \( 1 + 0.0194T + 97T^{2} \) |
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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
−8.021915476900336593482756204631, −7.40938273848264079437339743626, −6.57170120847441914632015377838, −5.78464161799476837380414490755, −4.72937850190224331406413743471, −3.87556241984402427231238466473, −3.09205466247752282732067794925, −2.72171306558403445201561688595, −1.23436648291945867257609904186, 0,
1.23436648291945867257609904186, 2.72171306558403445201561688595, 3.09205466247752282732067794925, 3.87556241984402427231238466473, 4.72937850190224331406413743471, 5.78464161799476837380414490755, 6.57170120847441914632015377838, 7.40938273848264079437339743626, 8.021915476900336593482756204631
