L(s) = 1 | + 2-s + 2.81·3-s + 4-s − 5-s + 2.81·6-s + 3.75·7-s + 8-s + 4.93·9-s − 10-s − 2.85·11-s + 2.81·12-s − 5.24·13-s + 3.75·14-s − 2.81·15-s + 16-s + 2.46·17-s + 4.93·18-s + 4.34·19-s − 20-s + 10.5·21-s − 2.85·22-s + 4.45·23-s + 2.81·24-s + 25-s − 5.24·26-s + 5.43·27-s + 3.75·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.62·3-s + 0.5·4-s − 0.447·5-s + 1.14·6-s + 1.41·7-s + 0.353·8-s + 1.64·9-s − 0.316·10-s − 0.861·11-s + 0.812·12-s − 1.45·13-s + 1.00·14-s − 0.727·15-s + 0.250·16-s + 0.596·17-s + 1.16·18-s + 0.996·19-s − 0.223·20-s + 2.30·21-s − 0.609·22-s + 0.928·23-s + 0.574·24-s + 0.200·25-s − 1.02·26-s + 1.04·27-s + 0.708·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.343303561\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.343303561\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 - 2.81T + 3T^{2} \) |
| 7 | \( 1 - 3.75T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 + 5.24T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 + 8.19T + 31T^{2} \) |
| 37 | \( 1 - 5.52T + 37T^{2} \) |
| 41 | \( 1 - 3.64T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 + 3.73T + 47T^{2} \) |
| 53 | \( 1 - 9.25T + 53T^{2} \) |
| 59 | \( 1 - 5.76T + 59T^{2} \) |
| 61 | \( 1 - 5.81T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 8.53T + 71T^{2} \) |
| 73 | \( 1 + 3.10T + 73T^{2} \) |
| 79 | \( 1 - 2.30T + 79T^{2} \) |
| 83 | \( 1 + 2.66T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.84238494367267721467942790676, −7.51512927424394303346191759940, −7.16492202909636656516062123733, −5.63191832117061736193296857580, −4.96234270629266625493770929190, −4.46128668878823919755030696884, −3.54457763907744987219753520396, −2.74772263010186333571642419669, −2.29293111038044849966129318070, −1.19440340527442960137637138238,
1.19440340527442960137637138238, 2.29293111038044849966129318070, 2.74772263010186333571642419669, 3.54457763907744987219753520396, 4.46128668878823919755030696884, 4.96234270629266625493770929190, 5.63191832117061736193296857580, 7.16492202909636656516062123733, 7.51512927424394303346191759940, 7.84238494367267721467942790676