| L(s) = 1 | + 2-s − 2·3-s + 4-s + 5-s − 2·6-s − 7-s + 8-s + 9-s + 10-s − 2·12-s − 14-s − 2·15-s + 16-s + 18-s − 2·19-s + 20-s + 2·21-s − 6·23-s − 2·24-s + 25-s + 4·27-s − 28-s + 6·29-s − 2·30-s − 8·31-s + 32-s − 35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.436·21-s − 1.25·23-s − 0.408·24-s + 1/5·25-s + 0.769·27-s − 0.188·28-s + 1.11·29-s − 0.365·30-s − 1.43·31-s + 0.176·32-s − 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−16.57059357708054, −16.04789666243778, −15.85765262755958, −14.78965188515105, −14.43640845349283, −13.84563499378508, −13.11077076361939, −12.62882257460345, −12.21332518288460, −11.57026635376368, −10.88126506310914, −10.68747994990887, −9.749611551012648, −9.371451470272665, −8.314760607263113, −7.719506735218142, −6.797555713270440, −6.295828331264029, −5.893080868778562, −5.291722792912102, −4.546006694879484, −3.955757152428367, −2.947125582049862, −2.217757285525949, −1.150593786998091, 0,
1.150593786998091, 2.217757285525949, 2.947125582049862, 3.955757152428367, 4.546006694879484, 5.291722792912102, 5.893080868778562, 6.295828331264029, 6.797555713270440, 7.719506735218142, 8.314760607263113, 9.371451470272665, 9.749611551012648, 10.68747994990887, 10.88126506310914, 11.57026635376368, 12.21332518288460, 12.62882257460345, 13.11077076361939, 13.84563499378508, 14.43640845349283, 14.78965188515105, 15.85765262755958, 16.04789666243778, 16.57059357708054
