L(s) = 1 | + (−0.951 − 1.30i)2-s + (−0.309 − 0.951i)3-s + (−0.500 + 1.53i)4-s + (−0.951 − 0.309i)5-s + (−0.951 + 1.30i)6-s + (−0.809 − 0.587i)7-s + (0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + (0.499 + 1.53i)10-s + (−0.587 − 0.190i)11-s + 1.61·12-s + (0.809 + 0.587i)13-s + 1.61i·14-s + 0.999i·15-s + (0.363 − 0.5i)17-s + (1.53 + 0.499i)18-s + ⋯ |
L(s) = 1 | + (−0.951 − 1.30i)2-s + (−0.309 − 0.951i)3-s + (−0.500 + 1.53i)4-s + (−0.951 − 0.309i)5-s + (−0.951 + 1.30i)6-s + (−0.809 − 0.587i)7-s + (0.951 − 0.309i)8-s + (−0.809 + 0.587i)9-s + (0.499 + 1.53i)10-s + (−0.587 − 0.190i)11-s + 1.61·12-s + (0.809 + 0.587i)13-s + 1.61i·14-s + 0.999i·15-s + (0.363 − 0.5i)17-s + (1.53 + 0.499i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09472167740\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09472167740\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 211 | \( 1 + (0.809 + 0.587i)T \) |
good | 2 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 11 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + 1.61T + T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
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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
−10.33118384676363256000864028863, −9.136160487413364408047098611631, −8.578712545250614003506487262374, −7.56536694804222088400507362827, −6.93443245193915815259938092980, −5.53076015300476129758409409629, −3.90665175574649274161024980520, −3.04199242463974218999989252205, −1.59174564648898276770946992627, −0.15209118831223742791347645294,
3.14362762540509556208084181619, 4.19844806842361299171218274205, 5.72917750467731670980410122139, 6.00508205727527462721263560845, 7.24242943046823131002994962952, 8.100076895885464279706001652146, 8.794121877682874254339482328145, 9.585680576052833534863196810606, 10.44324574274730308274733578560, 11.06291833365488716342549071906