| L(s) = 1 | − 5-s − 5·11-s + 13-s + 8·17-s + 4·19-s + 23-s + 25-s + 6·29-s + 31-s + 7·37-s + 7·41-s + 6·43-s + 3·47-s − 6·53-s + 5·55-s − 5·61-s − 65-s − 15·67-s − 6·71-s + 73-s + 17·79-s − 6·83-s − 8·85-s + 6·89-s − 4·95-s + 7·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.50·11-s + 0.277·13-s + 1.94·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.11·29-s + 0.179·31-s + 1.15·37-s + 1.09·41-s + 0.914·43-s + 0.437·47-s − 0.824·53-s + 0.674·55-s − 0.640·61-s − 0.124·65-s − 1.83·67-s − 0.712·71-s + 0.117·73-s + 1.91·79-s − 0.658·83-s − 0.867·85-s + 0.635·89-s − 0.410·95-s + 0.710·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.15849548603812, −12.56879780389019, −12.27335570128441, −11.90078858904634, −11.25632228056157, −10.81061687749441, −10.40312811488740, −9.968638718573004, −9.470384086865359, −8.992443474357481, −8.272726410844103, −7.898739930822933, −7.520170424154244, −7.323942612671647, −6.332016336939155, −5.934661298302070, −5.482850945914706, −4.912715415773581, −4.521630751591068, −3.785665751524438, −3.156077978910854, −2.861434176103611, −2.294194219728585, −1.161010943134853, −0.9633495151515883, 0,
0.9633495151515883, 1.161010943134853, 2.294194219728585, 2.861434176103611, 3.156077978910854, 3.785665751524438, 4.521630751591068, 4.912715415773581, 5.482850945914706, 5.934661298302070, 6.332016336939155, 7.323942612671647, 7.520170424154244, 7.898739930822933, 8.272726410844103, 8.992443474357481, 9.470384086865359, 9.968638718573004, 10.40312811488740, 10.81061687749441, 11.25632228056157, 11.90078858904634, 12.27335570128441, 12.56879780389019, 13.15849548603812
