| L(s) = 1 | − 2.77·2-s + 5.68·4-s + 2.28·5-s − 1.15·7-s − 10.2·8-s − 6.33·10-s − 5.37·11-s + 2.96·13-s + 3.19·14-s + 16.9·16-s + 3.70·17-s + 0.505·19-s + 13.0·20-s + 14.9·22-s + 7.09·23-s + 0.221·25-s − 8.21·26-s − 6.55·28-s + 29-s − 5.10·31-s − 26.6·32-s − 10.2·34-s − 2.63·35-s + 7.89·37-s − 1.40·38-s − 23.3·40-s + 3.69·41-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 2.84·4-s + 1.02·5-s − 0.435·7-s − 3.61·8-s − 2.00·10-s − 1.62·11-s + 0.821·13-s + 0.853·14-s + 4.24·16-s + 0.898·17-s + 0.115·19-s + 2.90·20-s + 3.17·22-s + 1.47·23-s + 0.0443·25-s − 1.61·26-s − 1.23·28-s + 0.185·29-s − 0.916·31-s − 4.71·32-s − 1.76·34-s − 0.444·35-s + 1.29·37-s − 0.227·38-s − 3.69·40-s + 0.577·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7109145682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7109145682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 0.505T + 19T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 - 7.89T + 37T^{2} \) |
| 41 | \( 1 - 3.69T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 - 7.73T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 7.09T + 59T^{2} \) |
| 61 | \( 1 - 0.599T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 8.69T + 73T^{2} \) |
| 79 | \( 1 + 5.08T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−10.18092857673370978014492182650, −9.437972462080472960077165725095, −8.802777489906141213772654415830, −7.83087629717116988379337659793, −7.20157071907832747527179245477, −6.06699302138059469757592235356, −5.52824150096782372185301712196, −3.11643486341628840164076397706, −2.27795072065360536853934638507, −0.932267702995663823057557968232,
0.932267702995663823057557968232, 2.27795072065360536853934638507, 3.11643486341628840164076397706, 5.52824150096782372185301712196, 6.06699302138059469757592235356, 7.20157071907832747527179245477, 7.83087629717116988379337659793, 8.802777489906141213772654415830, 9.437972462080472960077165725095, 10.18092857673370978014492182650
