| L(s) = 1 | + 0.415·2-s + 3-s − 1.82·4-s − 0.732·5-s + 0.415·6-s − 7-s − 1.59·8-s + 9-s − 0.304·10-s + 5.01·11-s − 1.82·12-s + 6.10·13-s − 0.415·14-s − 0.732·15-s + 2.99·16-s + 1.31·17-s + 0.415·18-s + 3.22·19-s + 1.33·20-s − 21-s + 2.08·22-s + 7.49·23-s − 1.59·24-s − 4.46·25-s + 2.53·26-s + 27-s + 1.82·28-s + ⋯ |
| L(s) = 1 | + 0.294·2-s + 0.577·3-s − 0.913·4-s − 0.327·5-s + 0.169·6-s − 0.377·7-s − 0.562·8-s + 0.333·9-s − 0.0963·10-s + 1.51·11-s − 0.527·12-s + 1.69·13-s − 0.111·14-s − 0.189·15-s + 0.748·16-s + 0.318·17-s + 0.0980·18-s + 0.739·19-s + 0.299·20-s − 0.218·21-s + 0.444·22-s + 1.56·23-s − 0.324·24-s − 0.892·25-s + 0.497·26-s + 0.192·27-s + 0.345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.714277106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.714277106\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 0.415T + 2T^{2} \) |
| 5 | \( 1 + 0.732T + 5T^{2} \) |
| 11 | \( 1 - 5.01T + 11T^{2} \) |
| 13 | \( 1 - 6.10T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 - 7.30T + 31T^{2} \) |
| 37 | \( 1 + 0.374T + 37T^{2} \) |
| 41 | \( 1 + 9.78T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 6.71T + 61T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 9.55T + 83T^{2} \) |
| 89 | \( 1 + 6.43T + 89T^{2} \) |
| 97 | \( 1 - 3.64T + 97T^{2} \) |
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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
−8.071910475471341708408781477738, −7.02879868089410402997849446364, −6.49465019807990214369818112790, −5.72066291519118369337586558491, −4.91715863995510210821206933293, −4.06113738640952143867619916017, −3.51113243969558370563823462127, −3.18730737729803733109275010794, −1.57686520110120742651871601171, −0.842789488289687137769959682035,
0.842789488289687137769959682035, 1.57686520110120742651871601171, 3.18730737729803733109275010794, 3.51113243969558370563823462127, 4.06113738640952143867619916017, 4.91715863995510210821206933293, 5.72066291519118369337586558491, 6.49465019807990214369818112790, 7.02879868089410402997849446364, 8.071910475471341708408781477738
