| L(s) = 1 | − 3.36·3-s + 0.576·7-s + 8.29·9-s + 11-s + 3.72·13-s − 6.51·17-s − 19-s − 1.93·21-s − 4.36·23-s − 17.8·27-s + 2.42·29-s − 2.15·31-s − 3.36·33-s + 3.21·37-s − 12.5·39-s + 5.93·41-s + 4.93·43-s − 12.6·47-s − 6.66·49-s + 21.8·51-s + 8.02·53-s + 3.36·57-s + 0.660·59-s − 10.4·61-s + 4.78·63-s + 5.87·67-s + 14.6·69-s + ⋯ |
| L(s) = 1 | − 1.94·3-s + 0.217·7-s + 2.76·9-s + 0.301·11-s + 1.03·13-s − 1.58·17-s − 0.229·19-s − 0.422·21-s − 0.909·23-s − 3.42·27-s + 0.450·29-s − 0.386·31-s − 0.585·33-s + 0.528·37-s − 2.00·39-s + 0.927·41-s + 0.753·43-s − 1.84·47-s − 0.952·49-s + 3.06·51-s + 1.10·53-s + 0.445·57-s + 0.0860·59-s − 1.33·61-s + 0.602·63-s + 0.717·67-s + 1.76·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 3.36T + 3T^{2} \) |
| 7 | \( 1 - 0.576T + 7T^{2} \) |
| 13 | \( 1 - 3.72T + 13T^{2} \) |
| 17 | \( 1 + 6.51T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 4.36T + 23T^{2} \) |
| 29 | \( 1 - 2.42T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 - 3.21T + 37T^{2} \) |
| 41 | \( 1 - 5.93T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 8.02T + 53T^{2} \) |
| 59 | \( 1 - 0.660T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 5.87T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 7.45T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.83589661672330049748423442186, −6.99183723167994755540760901583, −6.19363513780675348860031476919, −6.12076013953186846797784659814, −5.01686166503941793220868552915, −4.43161215763664818660334580906, −3.76430628065663320258191892349, −2.09209916746259551620164255363, −1.13993894243470798599645200060, 0,
1.13993894243470798599645200060, 2.09209916746259551620164255363, 3.76430628065663320258191892349, 4.43161215763664818660334580906, 5.01686166503941793220868552915, 6.12076013953186846797784659814, 6.19363513780675348860031476919, 6.99183723167994755540760901583, 7.83589661672330049748423442186
