| L(s) = 1 | − 2-s + 2.23·3-s + 4-s − 5-s − 2.23·6-s + 7-s − 8-s + 2.01·9-s + 10-s + 2.23·12-s − 5.12·13-s − 14-s − 2.23·15-s + 16-s + 6.65·17-s − 2.01·18-s + 1.15·19-s − 20-s + 2.23·21-s + 3.87·23-s − 2.23·24-s + 25-s + 5.12·26-s − 2.21·27-s + 28-s − 5.89·29-s + 2.23·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.447·5-s − 0.913·6-s + 0.377·7-s − 0.353·8-s + 0.670·9-s + 0.316·10-s + 0.646·12-s − 1.42·13-s − 0.267·14-s − 0.578·15-s + 0.250·16-s + 1.61·17-s − 0.474·18-s + 0.265·19-s − 0.223·20-s + 0.488·21-s + 0.808·23-s − 0.456·24-s + 0.200·25-s + 1.00·26-s − 0.425·27-s + 0.188·28-s − 1.09·29-s + 0.408·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.132098417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.132098417\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 6.65T + 17T^{2} \) |
| 19 | \( 1 - 1.15T + 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 - 8.71T + 47T^{2} \) |
| 53 | \( 1 - 8.28T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 - 2.54T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 6.86T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 0.406T + 97T^{2} \) |
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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
−7.71984854723614523070765886009, −7.53397141501571877998825919979, −6.86312704745298590597300028959, −5.60405823336353569706082901987, −5.08753476503080057334641745077, −3.94258648487291169376190434606, −3.33422129339134223644718132628, −2.57945538919811349799486136995, −1.88346431349118590049539035192, −0.73843343586857687839866900778,
0.73843343586857687839866900778, 1.88346431349118590049539035192, 2.57945538919811349799486136995, 3.33422129339134223644718132628, 3.94258648487291169376190434606, 5.08753476503080057334641745077, 5.60405823336353569706082901987, 6.86312704745298590597300028959, 7.53397141501571877998825919979, 7.71984854723614523070765886009
