L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.0809 + 1.73i)3-s − 1.00i·4-s + (0.662 + 2.13i)5-s + (1.16 + 1.28i)6-s + (2.89 + 2.89i)7-s + (−0.707 − 0.707i)8-s + (−2.98 − 0.280i)9-s + (1.97 + 1.04i)10-s + 1.62i·11-s + (1.73 + 0.0809i)12-s + (−1.74 + 1.74i)13-s + 4.09·14-s + (−3.74 + 0.972i)15-s − 1.00·16-s + (0.396 − 0.396i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.0467 + 0.998i)3-s − 0.500i·4-s + (0.296 + 0.955i)5-s + (0.476 + 0.522i)6-s + (1.09 + 1.09i)7-s + (−0.250 − 0.250i)8-s + (−0.995 − 0.0933i)9-s + (0.625 + 0.329i)10-s + 0.489i·11-s + (0.499 + 0.0233i)12-s + (−0.484 + 0.484i)13-s + 1.09·14-s + (−0.967 + 0.251i)15-s − 0.250·16-s + (0.0960 − 0.0960i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0220 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0220 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46515 + 1.49778i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46515 + 1.49778i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.0809 - 1.73i)T \) |
| 5 | \( 1 + (-0.662 - 2.13i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + (-2.89 - 2.89i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.62iT - 11T^{2} \) |
| 13 | \( 1 + (1.74 - 1.74i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.396 + 0.396i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.55iT - 19T^{2} \) |
| 23 | \( 1 + (0.0718 + 0.0718i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 37 | \( 1 + (-5.51 - 5.51i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.20iT - 41T^{2} \) |
| 43 | \( 1 + (-7.67 + 7.67i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.63 - 6.63i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.49 - 2.49i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.90T + 59T^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 + (-7.82 - 7.82i)T + 67iT^{2} \) |
| 71 | \( 1 - 7.38iT - 71T^{2} \) |
| 73 | \( 1 + (-7.37 + 7.37i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.229iT - 79T^{2} \) |
| 83 | \( 1 + (-0.833 - 0.833i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + (-7.96 - 7.96i)T + 97iT^{2} \) |
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
−10.41535090926127696504344602354, −9.517066504229579223414404623250, −9.002047340644769920005007600973, −7.81003317437574380208310422008, −6.65228275888965630304246631625, −5.60321740957419182404161060571, −4.97712635792501211186641844133, −4.06808026562878493572418154052, −2.78448861817075308950009781671, −2.12697066768794622343355707686,
0.855844462171780778080444945554, 2.02590027000122237560243905801, 3.65507407321466957042530567145, 4.74674271782059105327405722574, 5.52818379147827549619961567408, 6.32582203908034402582529385851, 7.58684962914850012273764615536, 7.85265886609692669692554710243, 8.617313684262882611657714227819, 9.769860475125372789632415074881