| L(s) = 1 | − 2-s + 3.08·3-s + 4-s + 2.06·5-s − 3.08·6-s − 1.37·7-s − 8-s + 6.51·9-s − 2.06·10-s + 11-s + 3.08·12-s − 6.97·13-s + 1.37·14-s + 6.35·15-s + 16-s − 7.76·17-s − 6.51·18-s + 2.06·20-s − 4.24·21-s − 22-s − 4.75·23-s − 3.08·24-s − 0.750·25-s + 6.97·26-s + 10.8·27-s − 1.37·28-s + 2.88·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.78·3-s + 0.5·4-s + 0.921·5-s − 1.25·6-s − 0.519·7-s − 0.353·8-s + 2.17·9-s − 0.651·10-s + 0.301·11-s + 0.890·12-s − 1.93·13-s + 0.367·14-s + 1.64·15-s + 0.250·16-s − 1.88·17-s − 1.53·18-s + 0.460·20-s − 0.925·21-s − 0.213·22-s − 0.991·23-s − 0.629·24-s − 0.150·25-s + 1.36·26-s + 2.08·27-s − 0.259·28-s + 0.536·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.08T + 3T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 7 | \( 1 + 1.37T + 7T^{2} \) |
| 13 | \( 1 + 6.97T + 13T^{2} \) |
| 17 | \( 1 + 7.76T + 17T^{2} \) |
| 23 | \( 1 + 4.75T + 23T^{2} \) |
| 29 | \( 1 - 2.88T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 + 1.44T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 + 7.97T + 43T^{2} \) |
| 47 | \( 1 + 1.12T + 47T^{2} \) |
| 53 | \( 1 - 1.30T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 - 1.03T + 61T^{2} \) |
| 67 | \( 1 + 3.41T + 67T^{2} \) |
| 71 | \( 1 + 0.748T + 71T^{2} \) |
| 73 | \( 1 - 6.61T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 1.50T + 89T^{2} \) |
| 97 | \( 1 + 8.39T + 97T^{2} \) |
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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
−7.61996339790306593943656230378, −6.92968603066950442181219797303, −6.55965167345581104033782334691, −5.42922822221518393801751083040, −4.46481990513625421890468587473, −3.70609667074966061193583942328, −2.68872852872093103386337259057, −2.24717775245705911175744766111, −1.74762575203388834046133123819, 0,
1.74762575203388834046133123819, 2.24717775245705911175744766111, 2.68872852872093103386337259057, 3.70609667074966061193583942328, 4.46481990513625421890468587473, 5.42922822221518393801751083040, 6.55965167345581104033782334691, 6.92968603066950442181219797303, 7.61996339790306593943656230378
