| L(s) = 1 | − 5-s − 11-s − 6·13-s − 5·17-s + 2·19-s + 23-s − 4·25-s + 4·29-s + 2·31-s + 41-s + 8·43-s − 47-s + 10·53-s + 55-s − 6·59-s − 7·61-s + 6·65-s + 3·67-s + 2·73-s + 79-s − 9·83-s + 5·85-s + 16·89-s − 2·95-s + 11·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.66·13-s − 1.21·17-s + 0.458·19-s + 0.208·23-s − 4/5·25-s + 0.742·29-s + 0.359·31-s + 0.156·41-s + 1.21·43-s − 0.145·47-s + 1.37·53-s + 0.134·55-s − 0.781·59-s − 0.896·61-s + 0.744·65-s + 0.366·67-s + 0.234·73-s + 0.112·79-s − 0.987·83-s + 0.542·85-s + 1.69·89-s − 0.205·95-s + 1.11·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−15.15680190150086, −14.59702221295087, −14.02794192450072, −13.57850332769418, −12.95532849361516, −12.40376668113846, −11.97562025190087, −11.49550396371264, −10.91617489149527, −10.27995955845994, −9.867204915002359, −9.197958972768217, −8.771895963312572, −7.998297775383294, −7.509695843090205, −7.145090408137683, −6.418250049830703, −5.803088426580496, −5.008948308017714, −4.615239716068067, −4.037100626987876, −3.173578247346227, −2.501827915647551, −2.019933865862560, −0.8104857914949024, 0,
0.8104857914949024, 2.019933865862560, 2.501827915647551, 3.173578247346227, 4.037100626987876, 4.615239716068067, 5.008948308017714, 5.803088426580496, 6.418250049830703, 7.145090408137683, 7.509695843090205, 7.998297775383294, 8.771895963312572, 9.197958972768217, 9.867204915002359, 10.27995955845994, 10.91617489149527, 11.49550396371264, 11.97562025190087, 12.40376668113846, 12.95532849361516, 13.57850332769418, 14.02794192450072, 14.59702221295087, 15.15680190150086
