| L(s) = 1 | + (1.22 − 1.22i)3-s − 2.44·5-s + (−1 − 2.44i)7-s − 2.99i·9-s − 2.44i·13-s + (−2.99 + 2.99i)15-s − 4.89·17-s + 2.44i·19-s + (−4.22 − 1.77i)21-s + 6i·23-s + 0.999·25-s + (−3.67 − 3.67i)27-s + 6i·29-s + (2.44 + 5.99i)35-s + 2·37-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)3-s − 1.09·5-s + (−0.377 − 0.925i)7-s − 0.999i·9-s − 0.679i·13-s + (−0.774 + 0.774i)15-s − 1.18·17-s + 0.561i·19-s + (−0.921 − 0.387i)21-s + 1.25i·23-s + 0.199·25-s + (−0.707 − 0.707i)27-s + 1.11i·29-s + (0.414 + 1.01i)35-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5031383937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5031383937\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 7 | \( 1 + (1 + 2.44i)T \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 12.2iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−9.033887201430420172441495189210, −8.107622245524328931143614204145, −7.61349618492408846948038710920, −6.96445439293984852176057128290, −6.09322676392984717018518930736, −4.66513689231864600778735069544, −3.68112011416872061149464306207, −3.15061004507011078291461059817, −1.60127924413133821402085607501, −0.18210282536723180763365338167,
2.24707519718873489754290335895, 3.02687637249399231725788108467, 4.24600916170197955468920684760, 4.57104469491077490045295172069, 5.93368740113836409080991762487, 6.89693733821419862029344013445, 7.85429780782177420565530793365, 8.623163836441373182564472221592, 9.078205626235991967468837685618, 9.893737882187780593124619736913
