| L(s) = 1 | − 2·5-s − 7-s + 2·11-s − 6·17-s − 4·19-s − 2·23-s − 25-s + 4·31-s + 2·35-s − 2·37-s − 2·41-s − 4·43-s + 49-s − 12·53-s − 4·55-s + 6·61-s + 12·67-s + 2·71-s + 2·73-s − 2·77-s + 4·83-s + 12·85-s − 2·89-s + 8·95-s + 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.603·11-s − 1.45·17-s − 0.917·19-s − 0.417·23-s − 1/5·25-s + 0.718·31-s + 0.338·35-s − 0.328·37-s − 0.312·41-s − 0.609·43-s + 1/7·49-s − 1.64·53-s − 0.539·55-s + 0.768·61-s + 1.46·67-s + 0.237·71-s + 0.234·73-s − 0.227·77-s + 0.439·83-s + 1.30·85-s − 0.211·89-s + 0.820·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.09363589185758, −13.73332374243826, −13.10264891063999, −12.69006404210267, −12.17806084402910, −11.69569168294612, −11.15534016975112, −10.94118543333462, −10.08332466620661, −9.777025955528659, −9.029838709781093, −8.642301079561254, −8.145969473728369, −7.666950998417640, −6.872498818810292, −6.585433616988950, −6.156890770002535, −5.324042544737523, −4.591413722657969, −4.253466857381393, −3.670634173945675, −3.136902936078159, −2.252221325098994, −1.785545950612300, −0.6793195979296898, 0,
0.6793195979296898, 1.785545950612300, 2.252221325098994, 3.136902936078159, 3.670634173945675, 4.253466857381393, 4.591413722657969, 5.324042544737523, 6.156890770002535, 6.585433616988950, 6.872498818810292, 7.666950998417640, 8.145969473728369, 8.642301079561254, 9.029838709781093, 9.777025955528659, 10.08332466620661, 10.94118543333462, 11.15534016975112, 11.69569168294612, 12.17806084402910, 12.69006404210267, 13.10264891063999, 13.73332374243826, 14.09363589185758
