| L(s) = 1 | − 5-s − 4·7-s − 3·9-s + 4·11-s − 2·13-s − 2·17-s + 4·19-s + 25-s − 2·29-s + 8·31-s + 4·35-s − 6·37-s − 6·41-s − 8·43-s + 3·45-s − 4·47-s + 9·49-s − 6·53-s − 4·55-s + 4·59-s + 2·61-s + 12·63-s + 2·65-s + 8·67-s − 6·73-s − 16·77-s + 9·81-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 9-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.676·35-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 0.447·45-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s + 0.520·59-s + 0.256·61-s + 1.51·63-s + 0.248·65-s + 0.977·67-s − 0.702·73-s − 1.82·77-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6190427942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6190427942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.59897370024931, −14.30510404612971, −13.59170500291267, −13.28530826966776, −12.55309048870522, −12.05078775475141, −11.62840233794379, −11.34206361011766, −10.37153799546936, −9.948832140748800, −9.459255669153475, −8.917987777487455, −8.466210330673902, −7.798293205481762, −6.994753978824983, −6.625581380185037, −6.246698915590742, −5.439374530011303, −4.881002353822976, −4.057435541394965, −3.369420165549608, −3.131381918489652, −2.315916364209912, −1.294638379837364, −0.2943545743529619,
0.2943545743529619, 1.294638379837364, 2.315916364209912, 3.131381918489652, 3.369420165549608, 4.057435541394965, 4.881002353822976, 5.439374530011303, 6.246698915590742, 6.625581380185037, 6.994753978824983, 7.798293205481762, 8.466210330673902, 8.917987777487455, 9.459255669153475, 9.948832140748800, 10.37153799546936, 11.34206361011766, 11.62840233794379, 12.05078775475141, 12.55309048870522, 13.28530826966776, 13.59170500291267, 14.30510404612971, 14.59897370024931
