| L(s) = 1 | − 2·5-s − 7-s − 3·9-s − 4·11-s + 2·13-s + 6·17-s + 8·19-s − 25-s + 6·29-s − 8·31-s + 2·35-s + 2·37-s + 2·41-s − 4·43-s + 6·45-s + 8·47-s + 49-s − 6·53-s + 8·55-s + 6·61-s + 3·63-s − 4·65-s − 4·67-s + 8·71-s + 10·73-s + 4·77-s + 16·79-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.338·35-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.894·45-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 0.768·61-s + 0.377·63-s − 0.496·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.458847434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458847434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.25833150477569, −13.81065602915357, −13.46037178302536, −12.67748626348057, −12.21535568295623, −11.87020414590909, −11.30605978230495, −10.79233530232111, −10.36553101070390, −9.534990873670684, −9.368616261034643, −8.419500337807462, −8.052468111704647, −7.650186204060547, −7.209025373827282, −6.349384917836006, −5.730063363071832, −5.298090440062183, −4.876775628858774, −3.725916884733717, −3.493099205744468, −2.941127566471665, −2.262978980400459, −1.091229737388023, −0.4780091889392700,
0.4780091889392700, 1.091229737388023, 2.262978980400459, 2.941127566471665, 3.493099205744468, 3.725916884733717, 4.876775628858774, 5.298090440062183, 5.730063363071832, 6.349384917836006, 7.209025373827282, 7.650186204060547, 8.052468111704647, 8.419500337807462, 9.368616261034643, 9.534990873670684, 10.36553101070390, 10.79233530232111, 11.30605978230495, 11.87020414590909, 12.21535568295623, 12.67748626348057, 13.46037178302536, 13.81065602915357, 14.25833150477569
