| L(s) = 1 | − 5-s + 7-s + 5·13-s − 4·17-s + 3·19-s + 25-s + 7·29-s + 7·31-s − 35-s + 8·37-s − 41-s + 43-s + 6·47-s − 6·49-s − 2·53-s − 5·61-s − 5·65-s − 11·67-s − 6·71-s − 5·73-s + 79-s − 8·83-s + 4·85-s + 12·89-s + 5·91-s − 3·95-s + 18·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.38·13-s − 0.970·17-s + 0.688·19-s + 1/5·25-s + 1.29·29-s + 1.25·31-s − 0.169·35-s + 1.31·37-s − 0.156·41-s + 0.152·43-s + 0.875·47-s − 6/7·49-s − 0.274·53-s − 0.640·61-s − 0.620·65-s − 1.34·67-s − 0.712·71-s − 0.585·73-s + 0.112·79-s − 0.878·83-s + 0.433·85-s + 1.27·89-s + 0.524·91-s − 0.307·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.73164178586594, −13.30606992966609, −12.97910619094594, −12.08482756582376, −11.91603967538010, −11.37078313186925, −10.87246918275938, −10.55994969484784, −9.903164375945134, −9.334794836769041, −8.726620099860325, −8.469025853065299, −7.878938221151320, −7.464879531515552, −6.749490255526610, −6.237675995281369, −5.948712579827937, −5.078678110653605, −4.529830553610001, −4.223937856430381, −3.480213583076884, −2.894416601507376, −2.363935036378213, −1.318792150427745, −1.050273420817640, 0,
1.050273420817640, 1.318792150427745, 2.363935036378213, 2.894416601507376, 3.480213583076884, 4.223937856430381, 4.529830553610001, 5.078678110653605, 5.948712579827937, 6.237675995281369, 6.749490255526610, 7.464879531515552, 7.878938221151320, 8.469025853065299, 8.726620099860325, 9.334794836769041, 9.903164375945134, 10.55994969484784, 10.87246918275938, 11.37078313186925, 11.91603967538010, 12.08482756582376, 12.97910619094594, 13.30606992966609, 13.73164178586594
