L(s) = 1 | + (0.707 + 0.707i)2-s + (0.273 + 1.71i)3-s + 1.00i·4-s + (1.75 + 1.75i)5-s + (−1.01 + 1.40i)6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (−2.85 + 0.936i)9-s + 2.47i·10-s + (2.12 − 2.12i)11-s + (−1.71 + 0.273i)12-s + (−3.47 + 0.950i)13-s + 1.00i·14-s + (−2.51 + 3.47i)15-s − 1.00·16-s + 3.03·17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.158 + 0.987i)3-s + 0.500i·4-s + (0.783 + 0.783i)5-s + (−0.414 + 0.572i)6-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s + (−0.950 + 0.312i)9-s + 0.783i·10-s + (0.640 − 0.640i)11-s + (−0.493 + 0.0790i)12-s + (−0.964 + 0.263i)13-s + 0.267i·14-s + (−0.649 + 0.897i)15-s − 0.250·16-s + 0.736·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.906338 + 1.99893i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.906338 + 1.99893i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.273 - 1.71i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + (3.47 - 0.950i)T \) |
good | 5 | \( 1 + (-1.75 - 1.75i)T + 5iT^{2} \) |
| 11 | \( 1 + (-2.12 + 2.12i)T - 11iT^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 + (-2.63 + 2.63i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.478T + 23T^{2} \) |
| 29 | \( 1 + 6.03iT - 29T^{2} \) |
| 31 | \( 1 + (2.07 - 2.07i)T - 31iT^{2} \) |
| 37 | \( 1 + (4.21 + 4.21i)T + 37iT^{2} \) |
| 41 | \( 1 + (-4.84 - 4.84i)T + 41iT^{2} \) |
| 43 | \( 1 - 4.59iT - 43T^{2} \) |
| 47 | \( 1 + (6.97 - 6.97i)T - 47iT^{2} \) |
| 53 | \( 1 + 5.90iT - 53T^{2} \) |
| 59 | \( 1 + (-3.76 + 3.76i)T - 59iT^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + (6.61 - 6.61i)T - 67iT^{2} \) |
| 71 | \( 1 + (-7.36 - 7.36i)T + 71iT^{2} \) |
| 73 | \( 1 + (8.06 + 8.06i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + (-9.96 - 9.96i)T + 83iT^{2} \) |
| 89 | \( 1 + (-10.9 + 10.9i)T - 89iT^{2} \) |
| 97 | \( 1 + (-10.7 + 10.7i)T - 97iT^{2} \) |
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
−11.21848903193634218701385114323, −10.04331210919159567118802120249, −9.508555316130504849420197515262, −8.527937741631253970737301461342, −7.44226483996035620046497530470, −6.33475886672815678975481125982, −5.55926546543059715843058296443, −4.64019880039813821482178673905, −3.41224535325974847141504422162, −2.47018629504328820434595295936,
1.19412377893285961555054917893, 2.10187063656373991202821718006, 3.52228068379574418248821934225, 5.01710138861203759814123118731, 5.62636100931359303159411638459, 6.85210817684914685381800019958, 7.66042044755198857398963746566, 8.848775210449949486413638238511, 9.615908354228565844542291439165, 10.46695188638480870230608038527