L(s) = 1 | − 0.482·2-s − 3-s − 1.76·4-s − 4.13·5-s + 0.482·6-s + 7-s + 1.81·8-s + 9-s + 1.99·10-s + 0.997·11-s + 1.76·12-s + 0.702·13-s − 0.482·14-s + 4.13·15-s + 2.65·16-s − 2.33·17-s − 0.482·18-s − 1.46·19-s + 7.30·20-s − 21-s − 0.480·22-s − 6.00·23-s − 1.81·24-s + 12.0·25-s − 0.338·26-s − 27-s − 1.76·28-s + ⋯ |
L(s) = 1 | − 0.340·2-s − 0.577·3-s − 0.883·4-s − 1.84·5-s + 0.196·6-s + 0.377·7-s + 0.642·8-s + 0.333·9-s + 0.630·10-s + 0.300·11-s + 0.510·12-s + 0.194·13-s − 0.128·14-s + 1.06·15-s + 0.664·16-s − 0.567·17-s − 0.113·18-s − 0.335·19-s + 1.63·20-s − 0.218·21-s − 0.102·22-s − 1.25·23-s − 0.370·24-s + 2.41·25-s − 0.0664·26-s − 0.192·27-s − 0.334·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2999299999\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2999299999\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
good | 2 | \( 1 + 0.482T + 2T^{2} \) |
| 5 | \( 1 + 4.13T + 5T^{2} \) |
| 11 | \( 1 - 0.997T + 11T^{2} \) |
| 13 | \( 1 - 0.702T + 13T^{2} \) |
| 17 | \( 1 + 2.33T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 + 6.00T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 - 9.76T + 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 + 2.43T + 41T^{2} \) |
| 43 | \( 1 + 9.17T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 0.531T + 53T^{2} \) |
| 59 | \( 1 - 9.50T + 59T^{2} \) |
| 61 | \( 1 - 0.836T + 61T^{2} \) |
| 67 | \( 1 + 6.93T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 8.28T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 7.66T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.303233406112296335192708014184, −7.978989806907190057907683653938, −7.10671051701562834300706926807, −6.39297701479476773605309499343, −5.21449872562438664331317238897, −4.53890949531422469870025705914, −4.05193421716587424668410676729, −3.30059337924916303089967015153, −1.58543111095270405576082874869, −0.35475248091521464898024614263,
0.35475248091521464898024614263, 1.58543111095270405576082874869, 3.30059337924916303089967015153, 4.05193421716587424668410676729, 4.53890949531422469870025705914, 5.21449872562438664331317238897, 6.39297701479476773605309499343, 7.10671051701562834300706926807, 7.978989806907190057907683653938, 8.303233406112296335192708014184