| L(s) = 1 | + 3·3-s − 8·4-s − 5·5-s + 9·9-s + 42·11-s − 24·12-s − 20·13-s − 15·15-s + 64·16-s − 66·17-s − 38·19-s + 40·20-s + 12·23-s + 25·25-s + 27·27-s − 258·29-s − 146·31-s + 126·33-s − 72·36-s + 434·37-s − 60·39-s + 282·41-s + 20·43-s − 336·44-s − 45·45-s + 72·47-s + 192·48-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 1/3·9-s + 1.15·11-s − 0.577·12-s − 0.426·13-s − 0.258·15-s + 16-s − 0.941·17-s − 0.458·19-s + 0.447·20-s + 0.108·23-s + 1/5·25-s + 0.192·27-s − 1.65·29-s − 0.845·31-s + 0.664·33-s − 1/3·36-s + 1.92·37-s − 0.246·39-s + 1.07·41-s + 0.0709·43-s − 1.15·44-s − 0.149·45-s + 0.223·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.627916200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.627916200\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 42 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 12 T + p^{3} T^{2} \) |
| 29 | \( 1 + 258 T + p^{3} T^{2} \) |
| 31 | \( 1 + 146 T + p^{3} T^{2} \) |
| 37 | \( 1 - 434 T + p^{3} T^{2} \) |
| 41 | \( 1 - 282 T + p^{3} T^{2} \) |
| 43 | \( 1 - 20 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 336 T + p^{3} T^{2} \) |
| 59 | \( 1 - 360 T + p^{3} T^{2} \) |
| 61 | \( 1 - 682 T + p^{3} T^{2} \) |
| 67 | \( 1 - 812 T + p^{3} T^{2} \) |
| 71 | \( 1 - 810 T + p^{3} T^{2} \) |
| 73 | \( 1 - 124 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1136 T + p^{3} T^{2} \) |
| 83 | \( 1 + 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1208 T + p^{3} T^{2} \) |
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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
−9.532932867269770097376798838076, −9.263903122118973854716304218953, −8.381209543380755691101969302575, −7.55987171739752666467652186944, −6.57376456620349579658919833311, −5.34506557078284385171397403857, −4.16496544846817890975825891837, −3.79583086141106652219899179560, −2.24735725431790433633378728544, −0.71505678122873496052001644949,
0.71505678122873496052001644949, 2.24735725431790433633378728544, 3.79583086141106652219899179560, 4.16496544846817890975825891837, 5.34506557078284385171397403857, 6.57376456620349579658919833311, 7.55987171739752666467652186944, 8.381209543380755691101969302575, 9.263903122118973854716304218953, 9.532932867269770097376798838076
