| L(s) = 1 | − 2.72·2-s + 5.43·4-s − 3.85·7-s − 9.37·8-s + 2.51·11-s − 1.31·13-s + 10.5·14-s + 14.7·16-s + 3.65·17-s − 2.96·19-s − 6.86·22-s + 5.72·23-s + 3.59·26-s − 20.9·28-s + 6.86·29-s − 31-s − 21.3·32-s − 9.97·34-s − 5.17·37-s + 8.09·38-s − 6.65·41-s − 6.41·43-s + 13.6·44-s − 15.6·46-s + 6.59·47-s + 7.87·49-s − 7.16·52-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.71·4-s − 1.45·7-s − 3.31·8-s + 0.759·11-s − 0.365·13-s + 2.81·14-s + 3.67·16-s + 0.887·17-s − 0.680·19-s − 1.46·22-s + 1.19·23-s + 0.704·26-s − 3.96·28-s + 1.27·29-s − 0.179·31-s − 3.77·32-s − 1.71·34-s − 0.850·37-s + 1.31·38-s − 1.03·41-s − 0.978·43-s + 2.06·44-s − 2.30·46-s + 0.962·47-s + 1.12·49-s − 0.993·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 - 2.51T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 29 | \( 1 - 6.86T + 29T^{2} \) |
| 37 | \( 1 + 5.17T + 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 - 6.59T + 47T^{2} \) |
| 53 | \( 1 + 9.83T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 + 7.16T + 73T^{2} \) |
| 79 | \( 1 - 5.97T + 79T^{2} \) |
| 83 | \( 1 + 4.35T + 83T^{2} \) |
| 89 | \( 1 - 0.353T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−7.56936237833124433513833195342, −7.12224180426500239151525873119, −6.36324316876976510173362165134, −6.14239229436076251113940718422, −4.85825516693976362525262248339, −3.35095732929177434330393239592, −3.08754927699352558474461983576, −1.95509037230588208631769617390, −0.997931172050911511457640423252, 0,
0.997931172050911511457640423252, 1.95509037230588208631769617390, 3.08754927699352558474461983576, 3.35095732929177434330393239592, 4.85825516693976362525262248339, 6.14239229436076251113940718422, 6.36324316876976510173362165134, 7.12224180426500239151525873119, 7.56936237833124433513833195342
