| L(s) = 1 | − i·5-s − 4·7-s + 3·9-s + 4i·11-s − 2i·13-s + 2·17-s − 4i·19-s + 4·23-s − 25-s − 2i·29-s + 8·31-s + 4i·35-s − 6i·37-s + 6·41-s − 8i·43-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 1.51·7-s + 9-s + 1.20i·11-s − 0.554i·13-s + 0.485·17-s − 0.917i·19-s + 0.834·23-s − 0.200·25-s − 0.371i·29-s + 1.43·31-s + 0.676i·35-s − 0.986i·37-s + 0.937·41-s − 1.21i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.427581995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427581995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.735294771613275697138642101526, −8.987203995116236834822125741331, −7.81045122553104118177974239225, −7.02739250018173297176270594124, −6.44690641588573265126419483500, −5.26677198803139651019079527498, −4.42047506778467486043921333025, −3.43963126844855277340650502455, −2.30841819569299581978980749726, −0.72994933058938776222132328125,
1.12674084242442291587603208345, 2.85511336048940103964150390171, 3.48431216676264882025228505285, 4.51059440765599261398820940446, 5.90031244472374950454902684352, 6.42643138239113102869084538153, 7.16963968563535773738178697267, 8.126689057602863281207603802207, 9.118613976944968813447542278707, 9.866639121820695831092662345320
