L(s) = 1 | − 2.51·2-s + 4.31·4-s − 2.91·5-s + 3.68·7-s − 5.81·8-s + 7.31·10-s − 1.04·11-s − 1.57·13-s − 9.25·14-s + 5.98·16-s + 4.12·17-s + 3.33·19-s − 12.5·20-s + 2.63·22-s + 4.23·23-s + 3.48·25-s + 3.96·26-s + 15.8·28-s − 8.74·29-s − 1.08·31-s − 3.40·32-s − 10.3·34-s − 10.7·35-s − 9.14·37-s − 8.38·38-s + 16.9·40-s + 3.00·41-s + ⋯ |
L(s) = 1 | − 1.77·2-s + 2.15·4-s − 1.30·5-s + 1.39·7-s − 2.05·8-s + 2.31·10-s − 0.316·11-s − 0.438·13-s − 2.47·14-s + 1.49·16-s + 0.999·17-s + 0.765·19-s − 2.81·20-s + 0.561·22-s + 0.882·23-s + 0.696·25-s + 0.778·26-s + 3.00·28-s − 1.62·29-s − 0.194·31-s − 0.602·32-s − 1.77·34-s − 1.81·35-s − 1.50·37-s − 1.36·38-s + 2.67·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 5 | \( 1 + 2.91T + 5T^{2} \) |
| 7 | \( 1 - 3.68T + 7T^{2} \) |
| 11 | \( 1 + 1.04T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 + 1.08T + 31T^{2} \) |
| 37 | \( 1 + 9.14T + 37T^{2} \) |
| 41 | \( 1 - 3.00T + 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 - 7.66T + 59T^{2} \) |
| 61 | \( 1 - 3.75T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 3.62T + 79T^{2} \) |
| 83 | \( 1 + 5.36T + 83T^{2} \) |
| 89 | \( 1 - 7.35T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−7.989856843212723293921108808692, −7.32998850698692634227924056709, −6.96512940478021386629317106599, −5.56056338646944796510496382047, −4.97752476949231923534464116152, −3.87285111130750633967064228497, −3.00111869609118906886445573266, −1.89017539998465142297548057077, −1.09747707099772490273208210547, 0,
1.09747707099772490273208210547, 1.89017539998465142297548057077, 3.00111869609118906886445573266, 3.87285111130750633967064228497, 4.97752476949231923534464116152, 5.56056338646944796510496382047, 6.96512940478021386629317106599, 7.32998850698692634227924056709, 7.989856843212723293921108808692