| L(s) = 1 | + 3·5-s − 3·7-s + 11-s − 2·13-s + 5·17-s + 19-s + 4·23-s + 4·25-s + 6·29-s − 2·31-s − 9·35-s + 8·37-s + 8·41-s + 13·43-s − 13·47-s + 2·49-s + 6·53-s + 3·55-s − 4·59-s − 13·61-s − 6·65-s + 4·67-s + 8·71-s − 3·73-s − 3·77-s − 4·79-s − 4·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s + 0.301·11-s − 0.554·13-s + 1.21·17-s + 0.229·19-s + 0.834·23-s + 4/5·25-s + 1.11·29-s − 0.359·31-s − 1.52·35-s + 1.31·37-s + 1.24·41-s + 1.98·43-s − 1.89·47-s + 2/7·49-s + 0.824·53-s + 0.404·55-s − 0.520·59-s − 1.66·61-s − 0.744·65-s + 0.488·67-s + 0.949·71-s − 0.351·73-s − 0.341·77-s − 0.450·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.984935456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984935456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.524548758129872633774664362571, −9.221900813441333316896786957028, −7.924107656862495380801900682255, −7.00612622598426407511060309816, −6.16799098038826968846498615960, −5.66548080355213698116568009078, −4.59164546840578345262367580791, −3.25222241798197103354320694490, −2.49847213158180922818846804858, −1.07996559647171411835733236564,
1.07996559647171411835733236564, 2.49847213158180922818846804858, 3.25222241798197103354320694490, 4.59164546840578345262367580791, 5.66548080355213698116568009078, 6.16799098038826968846498615960, 7.00612622598426407511060309816, 7.924107656862495380801900682255, 9.221900813441333316896786957028, 9.524548758129872633774664362571
