| L(s) = 1 | − 2·4-s + 5-s − 4·7-s − 3·11-s − 2·13-s + 4·16-s − 6·17-s − 2·20-s + 25-s + 8·28-s − 3·29-s + 7·31-s − 4·35-s − 8·37-s − 6·41-s − 4·43-s + 6·44-s − 6·47-s + 9·49-s + 4·52-s − 6·53-s − 3·55-s − 15·59-s + 5·61-s − 8·64-s − 2·65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 1.51·7-s − 0.904·11-s − 0.554·13-s + 16-s − 1.45·17-s − 0.447·20-s + 1/5·25-s + 1.51·28-s − 0.557·29-s + 1.25·31-s − 0.676·35-s − 1.31·37-s − 0.937·41-s − 0.609·43-s + 0.904·44-s − 0.875·47-s + 9/7·49-s + 0.554·52-s − 0.824·53-s − 0.404·55-s − 1.95·59-s + 0.640·61-s − 64-s − 0.248·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.56632611896340, −15.84489021353139, −15.44199610293729, −14.95234591070546, −13.96900344912967, −13.69556895497439, −13.20530000526345, −12.71244449233728, −12.37509453170336, −11.47117783613607, −10.66837217729375, −10.08270882424419, −9.783519736466002, −9.190506016905916, −8.620370611125498, −8.052672286828466, −7.145535631846718, −6.592271646650536, −6.022000586231617, −5.209001842139968, −4.760552971710474, −3.959278525319110, −3.144560563804680, −2.645225921828843, −1.573020251661425, 0, 0,
1.573020251661425, 2.645225921828843, 3.144560563804680, 3.959278525319110, 4.760552971710474, 5.209001842139968, 6.022000586231617, 6.592271646650536, 7.145535631846718, 8.052672286828466, 8.620370611125498, 9.190506016905916, 9.783519736466002, 10.08270882424419, 10.66837217729375, 11.47117783613607, 12.37509453170336, 12.71244449233728, 13.20530000526345, 13.69556895497439, 13.96900344912967, 14.95234591070546, 15.44199610293729, 15.84489021353139, 16.56632611896340
