| L(s) = 1 | + (−0.456 − 2.18i)5-s − 4.37i·7-s − 5.58i·11-s + 4.37·13-s − 5.58i·17-s − 4i·19-s + (−4.58 + 1.99i)25-s + 2.55i·29-s + 5.29·31-s + (−9.58 + 1.99i)35-s + 2.55·37-s − 6·41-s + 11.1·43-s + 6.92i·47-s − 12.1·49-s + ⋯ |
| L(s) = 1 | + (−0.204 − 0.978i)5-s − 1.65i·7-s − 1.68i·11-s + 1.21·13-s − 1.35i·17-s − 0.917i·19-s + (−0.916 + 0.399i)25-s + 0.473i·29-s + 0.950·31-s + (−1.61 + 0.338i)35-s + 0.419·37-s − 0.937·41-s + 1.70·43-s + 1.01i·47-s − 1.73·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.815354720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.815354720\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.456 + 2.18i)T \) |
| good | 7 | \( 1 + 4.37iT - 7T^{2} \) |
| 11 | \( 1 + 5.58iT - 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 + 5.58iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2.55iT - 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 - 2.55T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 7.84T + 53T^{2} \) |
| 59 | \( 1 + 1.58iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 3.16T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 - 7.16T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−8.525980493803668453005477839403, −7.75020411640424424823728788894, −7.02090899403470989867228818960, −6.16033384668599369084756275485, −5.30115243698929573984065493404, −4.41546224898159017159588494828, −3.77667000480995556430430043588, −2.89322621922759571752112452000, −0.986125826142088212858678477825, −0.74957084375389549208508397428,
1.79745197764304918431076453948, 2.38092701988575207112190162063, 3.50500550652651940758485545068, 4.26119717397400812254736935669, 5.42341585936148896782375986703, 6.16480731170555005450233549924, 6.60766508170205532963509886530, 7.72738927416224317909935678081, 8.273038204265486271257729692993, 9.049660710959814702065103260851
