| L(s) = 1 | − 2·2-s + 4·4-s + 1.61·5-s + 14.7·7-s − 8·8-s − 3.23·10-s − 50.9·13-s − 29.5·14-s + 16·16-s + 60.3·17-s − 26.4·19-s + 6.47·20-s + 29.1·23-s − 122.·25-s + 101.·26-s + 59.1·28-s + 150.·29-s + 29.3·31-s − 32·32-s − 120.·34-s + 23.9·35-s − 224.·37-s + 52.8·38-s − 12.9·40-s − 119.·41-s + 213.·43-s − 58.3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.144·5-s + 0.799·7-s − 0.353·8-s − 0.102·10-s − 1.08·13-s − 0.565·14-s + 0.250·16-s + 0.860·17-s − 0.319·19-s + 0.0723·20-s + 0.264·23-s − 0.979·25-s + 0.768·26-s + 0.399·28-s + 0.962·29-s + 0.170·31-s − 0.176·32-s − 0.608·34-s + 0.115·35-s − 0.998·37-s + 0.225·38-s − 0.0511·40-s − 0.456·41-s + 0.758·43-s − 0.186·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 1.61T + 125T^{2} \) |
| 7 | \( 1 - 14.7T + 343T^{2} \) |
| 13 | \( 1 + 50.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 60.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 29.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 150.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 29.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 224.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 119.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 175.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 63.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 708.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 151.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 31.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 65.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 230.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 618.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−8.292602331212229340146644230669, −7.65580935510828299232862037894, −7.01964928330586812502996073114, −6.00140976266927362824836126988, −5.18876460728886474127919326640, −4.34825128095928384942372338356, −3.10347893472541117242393468014, −2.14465890328972462217258793968, −1.23680493549568292920000755206, 0,
1.23680493549568292920000755206, 2.14465890328972462217258793968, 3.10347893472541117242393468014, 4.34825128095928384942372338356, 5.18876460728886474127919326640, 6.00140976266927362824836126988, 7.01964928330586812502996073114, 7.65580935510828299232862037894, 8.292602331212229340146644230669
