L(s) = 1 | − 2.42·2-s + 1.86·3-s + 3.87·4-s − 2.06·5-s − 4.51·6-s + 4.60·7-s − 4.55·8-s + 0.464·9-s + 5.00·10-s − 2.98·11-s + 7.22·12-s + 0.123·13-s − 11.1·14-s − 3.83·15-s + 3.29·16-s − 1.25·17-s − 1.12·18-s + 1.44·19-s − 8.00·20-s + 8.57·21-s + 7.23·22-s − 5.31·23-s − 8.48·24-s − 0.747·25-s − 0.298·26-s − 4.71·27-s + 17.8·28-s + ⋯ |
L(s) = 1 | − 1.71·2-s + 1.07·3-s + 1.93·4-s − 0.922·5-s − 1.84·6-s + 1.74·7-s − 1.61·8-s + 0.154·9-s + 1.58·10-s − 0.900·11-s + 2.08·12-s + 0.0341·13-s − 2.98·14-s − 0.991·15-s + 0.823·16-s − 0.304·17-s − 0.265·18-s + 0.330·19-s − 1.78·20-s + 1.87·21-s + 1.54·22-s − 1.10·23-s − 1.73·24-s − 0.149·25-s − 0.0585·26-s − 0.908·27-s + 3.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 3 | \( 1 - 1.86T + 3T^{2} \) |
| 5 | \( 1 + 2.06T + 5T^{2} \) |
| 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 + 2.98T + 11T^{2} \) |
| 13 | \( 1 - 0.123T + 13T^{2} \) |
| 17 | \( 1 + 1.25T + 17T^{2} \) |
| 19 | \( 1 - 1.44T + 19T^{2} \) |
| 23 | \( 1 + 5.31T + 23T^{2} \) |
| 29 | \( 1 + 0.662T + 29T^{2} \) |
| 31 | \( 1 + 9.01T + 31T^{2} \) |
| 37 | \( 1 + 6.93T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 47 | \( 1 + 0.503T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 + 8.70T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 + 8.40T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 0.0416T + 73T^{2} \) |
| 79 | \( 1 - 4.41T + 79T^{2} \) |
| 83 | \( 1 - 4.42T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 2.27T + 97T^{2} \) |
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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
−8.739212201557541746958344808291, −8.039870810598045446790932473167, −7.74242422909818549004135097656, −7.29852217596064930866544332819, −5.76257161621510902707905268848, −4.62512636440072946165337271103, −3.52936958569404546614695093516, −2.31309837328729992406842194288, −1.65848647868944561319118662962, 0,
1.65848647868944561319118662962, 2.31309837328729992406842194288, 3.52936958569404546614695093516, 4.62512636440072946165337271103, 5.76257161621510902707905268848, 7.29852217596064930866544332819, 7.74242422909818549004135097656, 8.039870810598045446790932473167, 8.739212201557541746958344808291