L(s) = 1 | − 2.71·2-s − 2.41·3-s + 5.34·4-s + 3.72·5-s + 6.54·6-s − 2.88·7-s − 9.06·8-s + 2.83·9-s − 10.1·10-s − 4.46·11-s − 12.9·12-s − 13-s + 7.82·14-s − 9.00·15-s + 13.8·16-s − 1.88·17-s − 7.68·18-s − 8.12·19-s + 19.9·20-s + 6.97·21-s + 12.1·22-s − 2.19·23-s + 21.9·24-s + 8.90·25-s + 2.71·26-s + 0.393·27-s − 15.4·28-s + ⋯ |
L(s) = 1 | − 1.91·2-s − 1.39·3-s + 2.67·4-s + 1.66·5-s + 2.67·6-s − 1.09·7-s − 3.20·8-s + 0.945·9-s − 3.19·10-s − 1.34·11-s − 3.72·12-s − 0.277·13-s + 2.09·14-s − 2.32·15-s + 3.47·16-s − 0.456·17-s − 1.81·18-s − 1.86·19-s + 4.45·20-s + 1.52·21-s + 2.58·22-s − 0.457·23-s + 4.47·24-s + 1.78·25-s + 0.531·26-s + 0.0756·27-s − 2.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2771981316\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2771981316\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 3.72T + 5T^{2} \) |
| 7 | \( 1 + 2.88T + 7T^{2} \) |
| 11 | \( 1 + 4.46T + 11T^{2} \) |
| 17 | \( 1 + 1.88T + 17T^{2} \) |
| 19 | \( 1 + 8.12T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 41 | \( 1 + 6.21T + 41T^{2} \) |
| 43 | \( 1 + 0.0964T + 43T^{2} \) |
| 47 | \( 1 - 4.49T + 47T^{2} \) |
| 53 | \( 1 - 1.45T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 - 1.81T + 61T^{2} \) |
| 67 | \( 1 - 7.20T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 3.27T + 73T^{2} \) |
| 79 | \( 1 + 2.69T + 79T^{2} \) |
| 83 | \( 1 + 8.82T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 3.79T + 97T^{2} \) |
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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
−10.02743440950984966504081814574, −8.913629377841977233703601596532, −8.297499843849515193098910852380, −6.84313627446899764534567427433, −6.52752685278881086240799078250, −5.94550831668923256427242466736, −5.04685936110969846585587967109, −2.73733969901207311257494525146, −2.02327004261953968119476024601, −0.51259419154245720296481269535,
0.51259419154245720296481269535, 2.02327004261953968119476024601, 2.73733969901207311257494525146, 5.04685936110969846585587967109, 5.94550831668923256427242466736, 6.52752685278881086240799078250, 6.84313627446899764534567427433, 8.297499843849515193098910852380, 8.913629377841977233703601596532, 10.02743440950984966504081814574