L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.499 + 0.866i)4-s + (−1.22 − 0.707i)11-s + (0.499 − 0.866i)16-s + (0.999 + 1.73i)22-s + (−1.22 + 0.707i)23-s + (−0.5 + 0.866i)25-s + (1.22 + 0.707i)29-s + (−1.22 + 0.707i)32-s − 1.41i·44-s + 2·46-s + (1.22 − 0.707i)50-s − 1.41i·53-s + (−0.999 − 1.73i)58-s + 0.999·64-s + ⋯ |
L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.499 + 0.866i)4-s + (−1.22 − 0.707i)11-s + (0.499 − 0.866i)16-s + (0.999 + 1.73i)22-s + (−1.22 + 0.707i)23-s + (−0.5 + 0.866i)25-s + (1.22 + 0.707i)29-s + (−1.22 + 0.707i)32-s − 1.41i·44-s + 2·46-s + (1.22 − 0.707i)50-s − 1.41i·53-s + (−0.999 − 1.73i)58-s + 0.999·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3458042308\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3458042308\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−8.671773423919688563987114516799, −8.215228286482958573368385204813, −7.65090125490298256478218353305, −6.77579408941512066982645212797, −5.62118330665145066102425839919, −5.20633441695031926336165519034, −3.87956755178082820975356621848, −2.96212317648039465925110848689, −2.20917855984096276975244082780, −1.12187769223755682377893871922,
0.32288522951787767770156516364, 1.85598555304591033844870426382, 2.78466678808260653226199162778, 4.09762569699919628649442285259, 4.83046088358181510458449318977, 5.94829828174593325504806339157, 6.43225277762716845408695125822, 7.33457114457574134180653407054, 7.960102840934641385327166826589, 8.278669562032410125822865670399