L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.518 − 1.65i)3-s − 1.00i·4-s + (−2.11 + 0.736i)5-s + (−1.53 − 0.801i)6-s + (−2.44 − 2.44i)7-s + (−0.707 − 0.707i)8-s + (−2.46 + 1.71i)9-s + (−0.972 + 2.01i)10-s + 5.62i·11-s + (−1.65 + 0.518i)12-s + (0.933 − 0.933i)13-s − 3.45·14-s + (2.31 + 3.10i)15-s − 1.00·16-s + (0.0532 − 0.0532i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.299 − 0.954i)3-s − 0.500i·4-s + (−0.944 + 0.329i)5-s + (−0.626 − 0.327i)6-s + (−0.922 − 0.922i)7-s + (−0.250 − 0.250i)8-s + (−0.820 + 0.571i)9-s + (−0.307 + 0.636i)10-s + 1.69i·11-s + (−0.477 + 0.149i)12-s + (0.259 − 0.259i)13-s − 0.922·14-s + (0.596 + 0.802i)15-s − 0.250·16-s + (0.0129 − 0.0129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.110549 + 0.168164i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.110549 + 0.168164i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.518 + 1.65i)T \) |
| 5 | \( 1 + (2.11 - 0.736i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (2.44 + 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 - 5.62iT - 11T^{2} \) |
| 13 | \( 1 + (-0.933 + 0.933i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.0532 + 0.0532i)T - 17iT^{2} \) |
| 23 | \( 1 + (0.841 + 0.841i)T + 23iT^{2} \) |
| 29 | \( 1 + 6.56T + 29T^{2} \) |
| 31 | \( 1 + 5.11T + 31T^{2} \) |
| 37 | \( 1 + (5.88 + 5.88i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.75iT - 41T^{2} \) |
| 43 | \( 1 + (7.57 - 7.57i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.64 + 2.64i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.67 + 9.67i)T + 53iT^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 + (-6.27 - 6.27i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.29iT - 71T^{2} \) |
| 73 | \( 1 + (7.35 - 7.35i)T - 73iT^{2} \) |
| 79 | \( 1 - 3.41iT - 79T^{2} \) |
| 83 | \( 1 + (-4.84 - 4.84i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.69T + 89T^{2} \) |
| 97 | \( 1 + (4.18 + 4.18i)T + 97iT^{2} \) |
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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
−10.41273875995143942018995841357, −9.502514229648751959914792817513, −8.076951092094181022851856715756, −7.08633944452114724242430761474, −6.84430067004457056477515306913, −5.45847581799265597612655084128, −4.20642088560751415592474497652, −3.31789042641333024668607757738, −1.87681484582221759990642830034, −0.096362737108088116924926828919,
3.20201058358360508648791181855, 3.62424147405396598077111937678, 4.92995170122649643828985205341, 5.78461845909586848227435210737, 6.48810544506822005162999939559, 7.907536969016331898724517945985, 8.866656477821851834187968233230, 9.216140120439140337492948879474, 10.66709295700003694252902649918, 11.45041778062252053630463635414