| L(s) = 1 | − 3-s + 5-s − 3.95i·7-s + 9-s + 2.16i·11-s − 6.98i·13-s − 15-s − 2.03·17-s + (−2.20 − 3.76i)19-s + 3.95i·21-s − 8.87i·23-s + 25-s − 27-s + 5.63i·29-s − 1.31·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.49i·7-s + 0.333·9-s + 0.653i·11-s − 1.93i·13-s − 0.258·15-s − 0.494·17-s + (−0.505 − 0.863i)19-s + 0.862i·21-s − 1.85i·23-s + 0.200·25-s − 0.192·27-s + 1.04i·29-s − 0.236·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8553173881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8553173881\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + (2.20 + 3.76i)T \) |
| good | 7 | \( 1 + 3.95iT - 7T^{2} \) |
| 11 | \( 1 - 2.16iT - 11T^{2} \) |
| 13 | \( 1 + 6.98iT - 13T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 23 | \( 1 + 8.87iT - 23T^{2} \) |
| 29 | \( 1 - 5.63iT - 29T^{2} \) |
| 31 | \( 1 + 1.31T + 31T^{2} \) |
| 37 | \( 1 + 1.77iT - 37T^{2} \) |
| 41 | \( 1 - 4.92iT - 41T^{2} \) |
| 43 | \( 1 + 9.57iT - 43T^{2} \) |
| 47 | \( 1 - 9.15iT - 47T^{2} \) |
| 53 | \( 1 - 5.41iT - 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 1.62T + 61T^{2} \) |
| 67 | \( 1 + 6.56T + 67T^{2} \) |
| 71 | \( 1 + 5.35T + 71T^{2} \) |
| 73 | \( 1 - 2.95T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 4.37iT - 83T^{2} \) |
| 89 | \( 1 - 6.91iT - 89T^{2} \) |
| 97 | \( 1 + 13.0iT - 97T^{2} \) |
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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
−7.79507521954541811849900994396, −7.17706239784926690941398846403, −6.57216003407697415676817230900, −5.82800203784801781398139921327, −4.80653798659313056685589055302, −4.49172525543714573822326853792, −3.37061006038496368335780833424, −2.42970159983223813770563573606, −1.12167734581721745009995037813, −0.26808568078045948933149337001,
1.65556293435415831872503169555, 2.13714517098447414705344395622, 3.35931774062068871424264502306, 4.30726205153654660455827653585, 5.14832069749559918072526945887, 5.95987171803917036276267358195, 6.19908868629988838208646217998, 7.07802572730853651246714083857, 8.061584684675553036616430629395, 8.841268686345153170551301125870
