| L(s) = 1 | − 3.46i·5-s + (2 + 1.73i)7-s − 3.46i·11-s − 3.46i·13-s − 2·19-s + 3.46i·23-s − 6.99·25-s − 6·29-s + 8·31-s + (5.99 − 6.92i)35-s − 2·37-s − 6.92i·41-s − 10.3i·43-s + (1.00 + 6.92i)49-s − 6·53-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s + (0.755 + 0.654i)7-s − 1.04i·11-s − 0.960i·13-s − 0.458·19-s + 0.722i·23-s − 1.39·25-s − 1.11·29-s + 1.43·31-s + (1.01 − 1.17i)35-s − 0.328·37-s − 1.08i·41-s − 1.58i·43-s + (0.142 + 0.989i)49-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516464003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516464003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−9.538757810048670710223571516680, −8.601687696210814962771173394443, −8.443688551158381539660317915026, −7.47724639276814183680221468456, −5.91276184414972441064452187670, −5.43450964107125108163893723856, −4.62942892014019535409008522987, −3.45392698246985011461660649199, −1.97226932497993461772976755695, −0.70507083385500222514702759654,
1.76129286182797138282354903230, 2.78389696342066649299941617996, 4.05998227547414099170188174269, 4.74257174425157228390661136450, 6.24622688126620880327132712726, 6.87160452710330688023772681563, 7.51192819520089582143565234675, 8.380163627580030627585190924310, 9.676006291914819441269344034236, 10.17717654675375195319275814981
