L(s) = 1 | + 2-s + 4-s − 5-s − 5.18i·7-s + 8-s − 10-s − 5.96·11-s + 5.86i·13-s − 5.18i·14-s + 16-s + 4.42i·17-s + 5.41·19-s − 20-s − 5.96·22-s + 2.69i·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.95i·7-s + 0.353·8-s − 0.316·10-s − 1.79·11-s + 1.62i·13-s − 1.38i·14-s + 0.250·16-s + 1.07i·17-s + 1.24·19-s − 0.223·20-s − 1.27·22-s + 0.562i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.280493053\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.280493053\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (-7.18 - 3.91i)T \) |
good | 7 | \( 1 + 5.18iT - 7T^{2} \) |
| 11 | \( 1 + 5.96T + 11T^{2} \) |
| 13 | \( 1 - 5.86iT - 13T^{2} \) |
| 17 | \( 1 - 4.42iT - 17T^{2} \) |
| 19 | \( 1 - 5.41T + 19T^{2} \) |
| 23 | \( 1 - 2.69iT - 23T^{2} \) |
| 29 | \( 1 - 6.15iT - 29T^{2} \) |
| 31 | \( 1 + 3.36iT - 31T^{2} \) |
| 37 | \( 1 - 4.86T + 37T^{2} \) |
| 41 | \( 1 - 4.68T + 41T^{2} \) |
| 43 | \( 1 + 12.5iT - 43T^{2} \) |
| 47 | \( 1 - 1.66iT - 47T^{2} \) |
| 53 | \( 1 - 0.782T + 53T^{2} \) |
| 59 | \( 1 - 5.16iT - 59T^{2} \) |
| 61 | \( 1 - 14.3iT - 61T^{2} \) |
| 71 | \( 1 + 5.99iT - 71T^{2} \) |
| 73 | \( 1 - 4.25T + 73T^{2} \) |
| 79 | \( 1 + 1.83iT - 79T^{2} \) |
| 83 | \( 1 - 9.73iT - 83T^{2} \) |
| 89 | \( 1 + 15.2iT - 89T^{2} \) |
| 97 | \( 1 - 6.02iT - 97T^{2} \) |
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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
−7.75247568430610395318885023584, −7.35043055431757162566621971891, −6.94128330676047171589904716075, −5.91185784527655211108104585272, −5.11029980326614985218134584238, −4.34385024536181618167573074436, −3.87758934368799026434922770608, −3.12294780132386922690648221857, −1.97214590049130125989603107398, −0.905843417216140484322585595997,
0.53957125186804389033477568960, 2.28739080528606285640982913150, 2.87106302039472300674140277209, 3.18917044977633453384533456162, 4.75963477271980128707185971996, 5.17647144684423637405971063077, 5.65505665489589298125694568811, 6.33083170949191001420957548095, 7.51535709837348073531968115246, 7.988762171600070466547652937154