| L(s) = 1 | − 11-s − 2·17-s − 8·19-s + 2·23-s − 5·25-s + 6·29-s − 2·37-s + 2·41-s + 4·43-s − 6·47-s + 8·53-s − 8·59-s + 4·61-s + 12·67-s + 10·71-s + 6·73-s − 10·79-s − 4·83-s + 10·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s − 0.485·17-s − 1.83·19-s + 0.417·23-s − 25-s + 1.11·29-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.875·47-s + 1.09·53-s − 1.04·59-s + 0.512·61-s + 1.46·67-s + 1.18·71-s + 0.702·73-s − 1.12·79-s − 0.439·83-s + 1.05·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.21212714435070, −14.43668816785592, −14.17907080929097, −13.41886302317108, −13.00826575236013, −12.55996727204425, −11.98767360083132, −11.33927560571778, −10.88899430496982, −10.36750409953218, −9.868811706061029, −9.217756887465105, −8.549198673960693, −8.294441725481513, −7.561946125050691, −6.918172726146146, −6.364143643034421, −5.922043013752583, −5.086268355989206, −4.562605906061022, −3.971359794972639, −3.289792552203068, −2.351434236083861, −2.047194874909873, −0.9093738188291490, 0,
0.9093738188291490, 2.047194874909873, 2.351434236083861, 3.289792552203068, 3.971359794972639, 4.562605906061022, 5.086268355989206, 5.922043013752583, 6.364143643034421, 6.918172726146146, 7.561946125050691, 8.294441725481513, 8.549198673960693, 9.217756887465105, 9.868811706061029, 10.36750409953218, 10.88899430496982, 11.33927560571778, 11.98767360083132, 12.55996727204425, 13.00826575236013, 13.41886302317108, 14.17907080929097, 14.43668816785592, 15.21212714435070
