| L(s) = 1 | − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s − 7-s − 8-s + 9-s + 2·10-s − 2·11-s − 2·12-s − 4·13-s + 14-s + 4·15-s + 16-s − 2·17-s − 18-s − 2·20-s + 2·21-s + 2·22-s + 23-s + 2·24-s − 25-s + 4·26-s + 4·27-s − 28-s − 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.447·20-s + 0.436·21-s + 0.426·22-s + 0.208·23-s + 0.408·24-s − 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.188·28-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.92382295269362, −12.59862959972212, −12.25774912299380, −11.66320417408224, −11.43549714071054, −10.97961314191745, −10.54900634067026, −10.05964625785880, −9.684332158202819, −9.086577934609112, −8.547576803884593, −8.108153584488661, −7.576042979566130, −7.137262099190202, −6.712483529097640, −6.325764909507342, −5.583429451159721, −5.098145183837446, −4.900918681370253, −4.091209079016595, −3.395502209582355, −3.083587853218567, −2.149861939960315, −1.768902821527471, −0.7171671107427035, 0, 0,
0.7171671107427035, 1.768902821527471, 2.149861939960315, 3.083587853218567, 3.395502209582355, 4.091209079016595, 4.900918681370253, 5.098145183837446, 5.583429451159721, 6.325764909507342, 6.712483529097640, 7.137262099190202, 7.576042979566130, 8.108153584488661, 8.547576803884593, 9.086577934609112, 9.684332158202819, 10.05964625785880, 10.54900634067026, 10.97961314191745, 11.43549714071054, 11.66320417408224, 12.25774912299380, 12.59862959972212, 12.92382295269362
