L(s) = 1 | − 1.13·2-s + 3.40·3-s − 0.701·4-s − 0.104·5-s − 3.88·6-s + 1.00·7-s + 3.07·8-s + 8.62·9-s + 0.118·10-s + 11-s − 2.38·12-s + 1.47·13-s − 1.14·14-s − 0.354·15-s − 2.10·16-s − 0.523·17-s − 9.82·18-s + 4.88·19-s + 0.0729·20-s + 3.42·21-s − 1.13·22-s − 3.18·23-s + 10.4·24-s − 4.98·25-s − 1.67·26-s + 19.1·27-s − 0.704·28-s + ⋯ |
L(s) = 1 | − 0.805·2-s + 1.96·3-s − 0.350·4-s − 0.0465·5-s − 1.58·6-s + 0.379·7-s + 1.08·8-s + 2.87·9-s + 0.0375·10-s + 0.301·11-s − 0.689·12-s + 0.407·13-s − 0.306·14-s − 0.0916·15-s − 0.526·16-s − 0.126·17-s − 2.31·18-s + 1.12·19-s + 0.0163·20-s + 0.747·21-s − 0.242·22-s − 0.664·23-s + 2.14·24-s − 0.997·25-s − 0.328·26-s + 3.68·27-s − 0.133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.261536586\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.261536586\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 2 | \( 1 + 1.13T + 2T^{2} \) |
| 3 | \( 1 - 3.40T + 3T^{2} \) |
| 5 | \( 1 + 0.104T + 5T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 + 0.523T + 17T^{2} \) |
| 19 | \( 1 - 4.88T + 19T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 + 7.69T + 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 + 4.28T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 + 4.44T + 53T^{2} \) |
| 59 | \( 1 - 3.97T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 7.92T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 1.05T + 83T^{2} \) |
| 89 | \( 1 + 9.63T + 89T^{2} \) |
| 97 | \( 1 + 3.03T + 97T^{2} \) |
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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
−9.460790774305062761429072717117, −8.694171022335246543088809236986, −8.075926197210236523570904808139, −7.64866106798029036339091251048, −6.74999408899245658731874014158, −5.12678217275312640216277672164, −4.07774449682831113423797590890, −3.48895013221028873257927561235, −2.17008746738491976649550398286, −1.28357232884682650816241017473,
1.28357232884682650816241017473, 2.17008746738491976649550398286, 3.48895013221028873257927561235, 4.07774449682831113423797590890, 5.12678217275312640216277672164, 6.74999408899245658731874014158, 7.64866106798029036339091251048, 8.075926197210236523570904808139, 8.694171022335246543088809236986, 9.460790774305062761429072717117