| L(s) = 1 | − 9·3-s + 96.3·5-s + 81·9-s − 179.·11-s − 56.8·13-s − 867.·15-s + 1.41e3·17-s − 1.29e3·19-s − 4.66e3·23-s + 6.16e3·25-s − 729·27-s − 5.00e3·29-s − 7.44e3·31-s + 1.61e3·33-s − 9.08e3·37-s + 511.·39-s + 6.62e3·41-s − 1.29e4·43-s + 7.80e3·45-s − 5.49e3·47-s − 1.27e4·51-s + 2.95e4·53-s − 1.73e4·55-s + 1.16e4·57-s + 1.05e3·59-s + 9.48e3·61-s − 5.48e3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.72·5-s + 0.333·9-s − 0.447·11-s − 0.0933·13-s − 0.995·15-s + 1.18·17-s − 0.821·19-s − 1.84·23-s + 1.97·25-s − 0.192·27-s − 1.10·29-s − 1.39·31-s + 0.258·33-s − 1.09·37-s + 0.0538·39-s + 0.615·41-s − 1.06·43-s + 0.574·45-s − 0.363·47-s − 0.686·51-s + 1.44·53-s − 0.771·55-s + 0.474·57-s + 0.0395·59-s + 0.326·61-s − 0.160·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 96.3T + 3.12e3T^{2} \) |
| 11 | \( 1 + 179.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 56.8T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.41e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.29e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.66e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.08e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.49e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.95e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.05e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.48e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.33e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.18e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.36e5T + 8.58e9T^{2} \) |
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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
−9.796135763238285245161693806181, −8.754216189562243225589175078607, −7.59684865624907748689837199291, −6.51696562881039861526819474097, −5.68533177585037145593370268854, −5.26228102428563177642512044190, −3.78251488475464816347125960107, −2.28030353764693032382853695290, −1.53445099053973242533941348184, 0,
1.53445099053973242533941348184, 2.28030353764693032382853695290, 3.78251488475464816347125960107, 5.26228102428563177642512044190, 5.68533177585037145593370268854, 6.51696562881039861526819474097, 7.59684865624907748689837199291, 8.754216189562243225589175078607, 9.796135763238285245161693806181
