L(s) = 1 | + (−1.10 − 0.884i)2-s + (0.434 + 1.95i)4-s + (−0.0402 − 0.0232i)5-s + (−1.97 + 1.76i)7-s + (1.24 − 2.53i)8-s + (0.0238 + 0.0612i)10-s + (−3.11 + 1.79i)11-s − 6.29i·13-s + (3.73 − 0.204i)14-s + (−3.62 + 1.69i)16-s + (−0.258 − 0.447i)17-s + (−2.80 − 1.62i)19-s + (0.0278 − 0.0886i)20-s + (5.02 + 0.772i)22-s + (3.47 − 6.01i)23-s + ⋯ |
L(s) = 1 | + (−0.780 − 0.625i)2-s + (0.217 + 0.976i)4-s + (−0.0179 − 0.0103i)5-s + (−0.744 + 0.667i)7-s + (0.441 − 0.897i)8-s + (0.00753 + 0.0193i)10-s + (−0.939 + 0.542i)11-s − 1.74i·13-s + (0.998 − 0.0547i)14-s + (−0.905 + 0.423i)16-s + (−0.0626 − 0.108i)17-s + (−0.644 − 0.371i)19-s + (0.00623 − 0.0198i)20-s + (1.07 + 0.164i)22-s + (0.724 − 1.25i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0571251 - 0.335766i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0571251 - 0.335766i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.10 + 0.884i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.97 - 1.76i)T \) |
good | 5 | \( 1 + (0.0402 + 0.0232i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.11 - 1.79i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6.29iT - 13T^{2} \) |
| 17 | \( 1 + (0.258 + 0.447i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.80 + 1.62i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.47 + 6.01i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.29iT - 29T^{2} \) |
| 31 | \( 1 + (1.05 + 1.82i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.12 + 0.650i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 9.12iT - 43T^{2} \) |
| 47 | \( 1 + (-2.32 + 4.02i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.91 - 2.83i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.42 + 0.825i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.10 + 0.637i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.72 - 3.88i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + (-5.57 - 9.64i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.75 - 4.76i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 8.25iT - 83T^{2} \) |
| 89 | \( 1 + (-7.38 + 12.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.16T + 97T^{2} \) |
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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
−10.28085056212147958812251701824, −9.925177576849529129265938262660, −8.698358391766017609133878617321, −8.118798083298377254518548909143, −7.07341444630536014058031976389, −5.95850807813160543330833905199, −4.66411945777542057085530486500, −3.11919406223699648370533612579, −2.40132552112395866969224277215, −0.24878325127874638586806851617,
1.73219817072024570203859461314, 3.49629016778335975879908664292, 4.90159724242411889417019339803, 6.02122969533826874833238941288, 6.92596126743763601046610482267, 7.57318409850200834897579391693, 8.758715860447679801229046264513, 9.386893553284877627316011048089, 10.33809197080609477037072829014, 10.99931884145248430154492966993