| L(s) = 1 | + 3.50·3-s − 6.74·5-s + 5.06·7-s − 14.7·9-s − 22.4·11-s + 20.4·13-s − 23.6·15-s + 45.3·17-s + 79.8·19-s + 17.7·21-s + 67.4·23-s − 79.4·25-s − 146.·27-s − 29·29-s − 133.·31-s − 78.6·33-s − 34.2·35-s + 343.·37-s + 71.6·39-s + 72.7·41-s − 232.·43-s + 99.3·45-s + 416.·47-s − 317.·49-s + 158.·51-s + 136.·53-s + 151.·55-s + ⋯ |
| L(s) = 1 | + 0.674·3-s − 0.603·5-s + 0.273·7-s − 0.545·9-s − 0.615·11-s + 0.436·13-s − 0.406·15-s + 0.647·17-s + 0.964·19-s + 0.184·21-s + 0.611·23-s − 0.635·25-s − 1.04·27-s − 0.185·29-s − 0.775·31-s − 0.414·33-s − 0.165·35-s + 1.52·37-s + 0.294·39-s + 0.277·41-s − 0.824·43-s + 0.329·45-s + 1.29·47-s − 0.925·49-s + 0.436·51-s + 0.354·53-s + 0.371·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 3.50T + 27T^{2} \) |
| 5 | \( 1 + 6.74T + 125T^{2} \) |
| 7 | \( 1 - 5.06T + 343T^{2} \) |
| 11 | \( 1 + 22.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 79.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 67.4T + 1.21e4T^{2} \) |
| 31 | \( 1 + 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 343.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 72.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 232.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 416.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 136.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 586.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 212.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 138.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 812.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 163.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 926.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 539.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 196.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.85e3T + 9.12e5T^{2} \) |
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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
−8.359020986656344765686052209440, −7.80177012506842109290970795814, −7.24918016950529542547621965259, −5.93577237413658637146304398178, −5.30668869523389195366637219334, −4.18381870989337841003654983175, −3.32304514795669270778682319522, −2.61774491647195512903151362471, −1.31847985339537015084684608111, 0,
1.31847985339537015084684608111, 2.61774491647195512903151362471, 3.32304514795669270778682319522, 4.18381870989337841003654983175, 5.30668869523389195366637219334, 5.93577237413658637146304398178, 7.24918016950529542547621965259, 7.80177012506842109290970795814, 8.359020986656344765686052209440
