| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 5·7-s + 9-s − 2·10-s + 2·11-s + 2·12-s + 10·14-s − 15-s − 4·16-s + 17-s + 2·18-s − 2·19-s − 2·20-s + 5·21-s + 4·22-s − 23-s + 25-s + 27-s + 10·28-s − 29-s − 2·30-s + 5·31-s − 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1.88·7-s + 1/3·9-s − 0.632·10-s + 0.603·11-s + 0.577·12-s + 2.67·14-s − 0.258·15-s − 16-s + 0.242·17-s + 0.471·18-s − 0.458·19-s − 0.447·20-s + 1.09·21-s + 0.852·22-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.88·28-s − 0.185·29-s − 0.365·30-s + 0.898·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.289946147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.289946147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.31034538738660, −13.92210279453963, −13.56488235453500, −12.85679422576884, −12.34981595598991, −11.91412680447439, −11.33784584689025, −11.21445929103246, −10.40662170414085, −9.774292289665728, −8.947092139675568, −8.576093589344664, −8.110665257454184, −7.466085173177534, −7.044961660641824, −6.273354063699676, −5.647695012752807, −5.154483915763490, −4.449869591740180, −4.177796415310029, −3.778000868000777, −2.775044961744110, −2.378384072030987, −1.585969600457511, −0.8228634769747323,
0.8228634769747323, 1.585969600457511, 2.378384072030987, 2.775044961744110, 3.778000868000777, 4.177796415310029, 4.449869591740180, 5.154483915763490, 5.647695012752807, 6.273354063699676, 7.044961660641824, 7.466085173177534, 8.110665257454184, 8.576093589344664, 8.947092139675568, 9.774292289665728, 10.40662170414085, 11.21445929103246, 11.33784584689025, 11.91412680447439, 12.34981595598991, 12.85679422576884, 13.56488235453500, 13.92210279453963, 14.31034538738660
