L(s) = 1 | + (−0.342 − 0.939i)3-s + (1.62 − 0.939i)7-s + (−0.766 + 0.642i)9-s + (0.326 + 0.118i)13-s + (0.866 − 0.5i)19-s + (−1.43 − 1.20i)21-s + (0.939 + 0.342i)25-s + (0.866 + 0.500i)27-s + (0.642 + 1.11i)31-s − 1.53·37-s − 0.347i·39-s + (0.223 − 1.26i)43-s + (1.26 − 2.19i)49-s + (−0.766 − 0.642i)57-s + (−1.93 + 0.342i)61-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)3-s + (1.62 − 0.939i)7-s + (−0.766 + 0.642i)9-s + (0.326 + 0.118i)13-s + (0.866 − 0.5i)19-s + (−1.43 − 1.20i)21-s + (0.939 + 0.342i)25-s + (0.866 + 0.500i)27-s + (0.642 + 1.11i)31-s − 1.53·37-s − 0.347i·39-s + (0.223 − 1.26i)43-s + (1.26 − 2.19i)49-s + (−0.766 − 0.642i)57-s + (−1.93 + 0.342i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.415822477\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.415822477\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.342 + 0.939i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (-0.642 - 1.11i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + 1.53T + T^{2} \) |
| 41 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.223 + 1.26i)T + (-0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (1.93 - 0.342i)T + (0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.20 - 1.43i)T + (-0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (0.642 - 0.233i)T + (0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (1.11 - 1.32i)T + (-0.173 - 0.984i)T^{2} \) |
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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
−8.472864166069145151660334925124, −7.68055597543754099142193342216, −7.15512837023383753748815701809, −6.59706513671005295835488573717, −5.31233460890075833203452014835, −5.06396033849884835100474401046, −4.02062477320528480210425712759, −2.87756545931564059546341388505, −1.67592815099659167448841468971, −1.05197060635659736507522986962,
1.34523521133633977032001947870, 2.52518126653566136103929665854, 3.48205578539595594774636685071, 4.54726955549720655909043740443, 5.00645565953653359881841481869, 5.70259077052126313468297635256, 6.39057756320527974912282617787, 7.65246431618940886241472735155, 8.237934895232964593134206864254, 8.876491907962873348812853807538