| L(s) = 1 | + 0.431i·3-s − i·5-s + 3.33·7-s + 2.81·9-s + 1.49·11-s + 5.60·13-s + 0.431·15-s + 0.587i·17-s − 0.133·19-s + 1.43i·21-s + (−3.96 − 2.70i)23-s − 25-s + 2.51i·27-s − 1.54·29-s + 1.79i·31-s + ⋯ |
| L(s) = 1 | + 0.249i·3-s − 0.447i·5-s + 1.25·7-s + 0.937·9-s + 0.450·11-s + 1.55·13-s + 0.111·15-s + 0.142i·17-s − 0.0306·19-s + 0.313i·21-s + (−0.825 − 0.563i)23-s − 0.200·25-s + 0.483i·27-s − 0.286·29-s + 0.321i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.406615887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.406615887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (3.96 + 2.70i)T \) |
| good | 3 | \( 1 - 0.431iT - 3T^{2} \) |
| 7 | \( 1 - 3.33T + 7T^{2} \) |
| 11 | \( 1 - 1.49T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 - 0.587iT - 17T^{2} \) |
| 19 | \( 1 + 0.133T + 19T^{2} \) |
| 29 | \( 1 + 1.54T + 29T^{2} \) |
| 31 | \( 1 - 1.79iT - 31T^{2} \) |
| 37 | \( 1 + 5.62iT - 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 - 6.73iT - 47T^{2} \) |
| 53 | \( 1 - 0.671iT - 53T^{2} \) |
| 59 | \( 1 - 3.46iT - 59T^{2} \) |
| 61 | \( 1 + 13.7iT - 61T^{2} \) |
| 67 | \( 1 + 3.10T + 67T^{2} \) |
| 71 | \( 1 - 12.2iT - 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 8.78T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 1.49iT - 89T^{2} \) |
| 97 | \( 1 - 1.91iT - 97T^{2} \) |
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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
−9.157337686906898009915627015471, −8.399608635161993567506458683051, −7.905749185151609685383000813563, −6.85347632408100775765523192874, −5.99539833130886382790199943457, −5.04804903725924076269489002052, −4.27175017961175378099472043090, −3.64074041444168125748966286940, −1.92195068215947565763987662629, −1.17478070726360707516659797957,
1.26326560634970766775120062673, 1.97107064334966130617944287054, 3.51039708211873336282279689368, 4.21364634116098491235452694911, 5.19673032795498860304623109137, 6.19350218019064247276658154103, 6.86887560412256270000453624580, 7.80103872299207565907126899334, 8.290383514234161701568202931286, 9.201278687030105145812838577881
