| L(s) = 1 | − 3-s − 0.787i·5-s + i·7-s + 9-s + 4.29i·11-s + (2.55 + 2.54i)13-s + 0.787i·15-s + 1.31·17-s − 6.60i·19-s − i·21-s + 0.194·23-s + 4.38·25-s − 27-s + 3.31·29-s − 4.29i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.352i·5-s + 0.377i·7-s + 0.333·9-s + 1.29i·11-s + (0.709 + 0.704i)13-s + 0.203i·15-s + 0.318·17-s − 1.51i·19-s − 0.218i·21-s + 0.0405·23-s + 0.876·25-s − 0.192·27-s + 0.614·29-s − 0.747i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.556283984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.556283984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-2.55 - 2.54i)T \) |
| good | 5 | \( 1 + 0.787iT - 5T^{2} \) |
| 11 | \( 1 - 4.29iT - 11T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 + 6.60iT - 19T^{2} \) |
| 23 | \( 1 - 0.194T + 23T^{2} \) |
| 29 | \( 1 - 3.31T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 6.88iT - 37T^{2} \) |
| 41 | \( 1 + 4.06iT - 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 - 7.60iT - 47T^{2} \) |
| 53 | \( 1 - 2.27T + 53T^{2} \) |
| 59 | \( 1 - 5.88iT - 59T^{2} \) |
| 61 | \( 1 + 9.68T + 61T^{2} \) |
| 67 | \( 1 - 15.0iT - 67T^{2} \) |
| 71 | \( 1 - 10.5iT - 71T^{2} \) |
| 73 | \( 1 + 2.47iT - 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 14.0iT - 83T^{2} \) |
| 89 | \( 1 - 3.08iT - 89T^{2} \) |
| 97 | \( 1 - 5.54iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.806405039095805968198728303858, −7.47987094249995812361596583421, −7.05051537194135288291311771403, −6.26901801137007703954009075453, −5.47874226129104773099969039113, −4.65269728983142642173955193906, −4.25667231485336674978701786632, −2.94665854038689754576037996618, −1.97440058952634146309203540343, −0.927149465971928689314678188825,
0.62180665251365745110686671398, 1.56140976261312273524554609359, 3.13837403647358868322183525148, 3.47977025449579621755842373617, 4.59112804245536442412182847239, 5.46795794132563737478261737355, 6.13439693246570324957650008482, 6.58665497506943751979797990747, 7.63628444892485563938073560218, 8.217697077699348773611936272328
