| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 1.35·7-s − 8-s + 9-s + 10-s + 3.19·11-s − 12-s − 1.35·14-s + 15-s + 16-s − 2.04·17-s − 18-s − 5.34·19-s − 20-s − 1.35·21-s − 3.19·22-s − 8.32·23-s + 24-s + 25-s − 27-s + 1.35·28-s + 8.07·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.512·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.964·11-s − 0.288·12-s − 0.362·14-s + 0.258·15-s + 0.250·16-s − 0.496·17-s − 0.235·18-s − 1.22·19-s − 0.223·20-s − 0.296·21-s − 0.681·22-s − 1.73·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.256·28-s + 1.49·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9416862975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9416862975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 + 5.34T + 19T^{2} \) |
| 23 | \( 1 + 8.32T + 23T^{2} \) |
| 29 | \( 1 - 8.07T + 29T^{2} \) |
| 31 | \( 1 + 0.185T + 31T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 2.91T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 5.74T + 53T^{2} \) |
| 59 | \( 1 + 1.62T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 - 1.55T + 67T^{2} \) |
| 71 | \( 1 + 2.32T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 + 1.42T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−8.253833641425252630480857580540, −7.62835451793285502448403145515, −6.71307004264513881874196637460, −6.31917177135835036275046433639, −5.46451343059103464595566188802, −4.27401239050428083200440962019, −4.06629924642997500877030738959, −2.59255091548729693927150326837, −1.69617673626020252369096680466, −0.61614507003583976689337394429,
0.61614507003583976689337394429, 1.69617673626020252369096680466, 2.59255091548729693927150326837, 4.06629924642997500877030738959, 4.27401239050428083200440962019, 5.46451343059103464595566188802, 6.31917177135835036275046433639, 6.71307004264513881874196637460, 7.62835451793285502448403145515, 8.253833641425252630480857580540
