| L(s) = 1 | + 1.43·2-s − 1.34·3-s + 0.0487·4-s + 5-s − 1.92·6-s + 0.955·7-s − 2.79·8-s − 1.19·9-s + 1.43·10-s − 1.86·11-s − 0.0654·12-s + 0.938·13-s + 1.36·14-s − 1.34·15-s − 4.09·16-s + 17-s − 1.71·18-s + 1.75·19-s + 0.0487·20-s − 1.28·21-s − 2.66·22-s + 1.42·23-s + 3.74·24-s + 25-s + 1.34·26-s + 5.63·27-s + 0.0465·28-s + ⋯ |
| L(s) = 1 | + 1.01·2-s − 0.775·3-s + 0.0243·4-s + 0.447·5-s − 0.784·6-s + 0.361·7-s − 0.987·8-s − 0.399·9-s + 0.452·10-s − 0.561·11-s − 0.0188·12-s + 0.260·13-s + 0.365·14-s − 0.346·15-s − 1.02·16-s + 0.242·17-s − 0.403·18-s + 0.402·19-s + 0.0109·20-s − 0.280·21-s − 0.568·22-s + 0.296·23-s + 0.765·24-s + 0.200·25-s + 0.263·26-s + 1.08·27-s + 0.00880·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 3 | \( 1 + 1.34T + 3T^{2} \) |
| 7 | \( 1 - 0.955T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 - 0.938T + 13T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 23 | \( 1 - 1.42T + 23T^{2} \) |
| 29 | \( 1 - 8.14T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 + 0.116T + 37T^{2} \) |
| 41 | \( 1 - 1.63T + 41T^{2} \) |
| 43 | \( 1 - 1.87T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 + 7.88T + 53T^{2} \) |
| 59 | \( 1 + 8.85T + 59T^{2} \) |
| 61 | \( 1 + 6.46T + 61T^{2} \) |
| 67 | \( 1 + 1.74T + 67T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 1.99T + 89T^{2} \) |
| 97 | \( 1 + 9.15T + 97T^{2} \) |
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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
−7.64407549480581197178860313747, −6.60530021756898180331380804861, −6.08412911404303326390697470583, −5.45967678579876027841594227957, −4.94727399949454963843997917919, −4.34698005495189697004224119346, −3.20259941551098838775664939893, −2.68250013847122564951224679503, −1.29879275176086460368260706382, 0,
1.29879275176086460368260706382, 2.68250013847122564951224679503, 3.20259941551098838775664939893, 4.34698005495189697004224119346, 4.94727399949454963843997917919, 5.45967678579876027841594227957, 6.08412911404303326390697470583, 6.60530021756898180331380804861, 7.64407549480581197178860313747
