| L(s) = 1 | − 2-s + 2.61·3-s + 4-s − 5-s − 2.61·6-s − 7-s − 8-s + 3.85·9-s + 10-s + 2.61·12-s − 2·13-s + 14-s − 2.61·15-s + 16-s + 1.61·17-s − 3.85·18-s − 6.85·19-s − 20-s − 2.61·21-s + 6·23-s − 2.61·24-s + 25-s + 2·26-s + 2.23·27-s − 28-s − 3.23·29-s + 2.61·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.51·3-s + 0.5·4-s − 0.447·5-s − 1.06·6-s − 0.377·7-s − 0.353·8-s + 1.28·9-s + 0.316·10-s + 0.755·12-s − 0.554·13-s + 0.267·14-s − 0.675·15-s + 0.250·16-s + 0.392·17-s − 0.908·18-s − 1.57·19-s − 0.223·20-s − 0.571·21-s + 1.25·23-s − 0.534·24-s + 0.200·25-s + 0.392·26-s + 0.430·27-s − 0.188·28-s − 0.600·29-s + 0.477·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 6.85T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 + 6.47T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 + 7.61T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 + 9.70T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 8.47T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 9.56T + 97T^{2} \) |
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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
−7.46264429169674004532743692244, −7.21189559697520536471377097697, −6.34922902067241816381709942199, −5.38928741433899459529737440625, −4.30819419361542607104746686627, −3.71539330290363594427277486396, −2.84849362557972434686489115053, −2.38245051820441006457868111340, −1.37022022837728817710405307204, 0,
1.37022022837728817710405307204, 2.38245051820441006457868111340, 2.84849362557972434686489115053, 3.71539330290363594427277486396, 4.30819419361542607104746686627, 5.38928741433899459529737440625, 6.34922902067241816381709942199, 7.21189559697520536471377097697, 7.46264429169674004532743692244
