| L(s) = 1 | + 4.30·3-s + 28.3·7-s − 8.49·9-s − 65.2·11-s − 33.6·13-s + 73.3·17-s − 134.·19-s + 121.·21-s − 14.7·23-s − 152.·27-s − 224.·29-s − 68.8·31-s − 280.·33-s − 196.·37-s − 144.·39-s − 143.·41-s + 15.0·43-s + 134.·47-s + 458.·49-s + 315.·51-s + 262.·53-s − 576.·57-s + 119.·59-s + 16.5·61-s − 240.·63-s + 545.·67-s − 63.2·69-s + ⋯ |
| L(s) = 1 | + 0.827·3-s + 1.52·7-s − 0.314·9-s − 1.78·11-s − 0.718·13-s + 1.04·17-s − 1.61·19-s + 1.26·21-s − 0.133·23-s − 1.08·27-s − 1.43·29-s − 0.398·31-s − 1.48·33-s − 0.872·37-s − 0.594·39-s − 0.545·41-s + 0.0534·43-s + 0.417·47-s + 1.33·49-s + 0.865·51-s + 0.681·53-s − 1.34·57-s + 0.264·59-s + 0.0347·61-s − 0.481·63-s + 0.994·67-s − 0.110·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4.30T + 27T^{2} \) |
| 7 | \( 1 - 28.3T + 343T^{2} \) |
| 11 | \( 1 + 65.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 14.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 68.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 143.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 15.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 262.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 119.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 16.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 545.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 199.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 43.2T + 3.89e5T^{2} \) |
| 79 | \( 1 + 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 723.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3T + 9.12e5T^{2} \) |
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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
−9.326296780481519859541707548128, −8.251587048414994800227621507297, −8.054009203149407344303178634365, −7.22539913551722082217566803297, −5.56730194099167278241759052525, −5.08321933327784314425046074035, −3.85772264252258717715249302470, −2.57115108947149486573428068472, −1.90195662217019154839614845305, 0,
1.90195662217019154839614845305, 2.57115108947149486573428068472, 3.85772264252258717715249302470, 5.08321933327784314425046074035, 5.56730194099167278241759052525, 7.22539913551722082217566803297, 8.054009203149407344303178634365, 8.251587048414994800227621507297, 9.326296780481519859541707548128
