| L(s) = 1 | + 0.0344·2-s + 3-s − 1.99·4-s + 4.08·5-s + 0.0344·6-s − 0.0879·7-s − 0.137·8-s + 9-s + 0.140·10-s + 2.81·11-s − 1.99·12-s − 6.11·13-s − 0.00303·14-s + 4.08·15-s + 3.99·16-s + 0.723·17-s + 0.0344·18-s + 7.09·19-s − 8.16·20-s − 0.0879·21-s + 0.0969·22-s − 6.69·23-s − 0.137·24-s + 11.6·25-s − 0.210·26-s + 27-s + 0.175·28-s + ⋯ |
| L(s) = 1 | + 0.0243·2-s + 0.577·3-s − 0.999·4-s + 1.82·5-s + 0.0140·6-s − 0.0332·7-s − 0.0487·8-s + 0.333·9-s + 0.0445·10-s + 0.847·11-s − 0.577·12-s − 1.69·13-s − 0.000811·14-s + 1.05·15-s + 0.998·16-s + 0.175·17-s + 0.00813·18-s + 1.62·19-s − 1.82·20-s − 0.0191·21-s + 0.0206·22-s − 1.39·23-s − 0.0281·24-s + 2.33·25-s − 0.0413·26-s + 0.192·27-s + 0.0332·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.686235498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.686235498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0344T + 2T^{2} \) |
| 5 | \( 1 - 4.08T + 5T^{2} \) |
| 7 | \( 1 + 0.0879T + 7T^{2} \) |
| 11 | \( 1 - 2.81T + 11T^{2} \) |
| 13 | \( 1 + 6.11T + 13T^{2} \) |
| 17 | \( 1 - 0.723T + 17T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 23 | \( 1 + 6.69T + 23T^{2} \) |
| 29 | \( 1 - 2.45T + 29T^{2} \) |
| 31 | \( 1 - 1.49T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 1.90T + 41T^{2} \) |
| 43 | \( 1 + 0.573T + 43T^{2} \) |
| 47 | \( 1 + 8.41T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 3.34T + 59T^{2} \) |
| 61 | \( 1 + 2.33T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 + 6.73T + 73T^{2} \) |
| 79 | \( 1 - 2.98T + 79T^{2} \) |
| 83 | \( 1 + 7.75T + 83T^{2} \) |
| 89 | \( 1 + 1.35T + 89T^{2} \) |
| 97 | \( 1 - 7.69T + 97T^{2} \) |
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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
−11.98427709773554378798088924298, −10.13209822134002116797906512208, −9.722566638794428429013407439768, −9.239842798485422775344387605960, −8.051411603562587521436144390836, −6.77237720880125830628557985491, −5.56098230595533337849951791360, −4.74197477740014207165809648290, −3.13904709390568442311780919825, −1.68818962241488991577358890318,
1.68818962241488991577358890318, 3.13904709390568442311780919825, 4.74197477740014207165809648290, 5.56098230595533337849951791360, 6.77237720880125830628557985491, 8.051411603562587521436144390836, 9.239842798485422775344387605960, 9.722566638794428429013407439768, 10.13209822134002116797906512208, 11.98427709773554378798088924298
