| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2.60·7-s + 8-s + 9-s + 10-s − 12-s − 2.60·14-s − 15-s + 16-s + 2.60·17-s + 18-s + 2.60·19-s + 20-s + 2.60·21-s + 8.60·23-s − 24-s + 25-s − 27-s − 2.60·28-s − 2.60·29-s − 30-s + 6·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.984·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.288·12-s − 0.696·14-s − 0.258·15-s + 0.250·16-s + 0.631·17-s + 0.235·18-s + 0.597·19-s + 0.223·20-s + 0.568·21-s + 1.79·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.492·28-s − 0.483·29-s − 0.182·30-s + 1.07·31-s + 0.176·32-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
| \(L(1)\) |
\(\approx\) |
\(2.569241998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.569241998\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 - 8.60T + 23T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 - 8.60T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + 0.788T + 89T^{2} \) |
| 97 | \( 1 - 8.60T + 97T^{2} \) |
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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
−8.130850228679215748899614826416, −7.11961595923255558568530422163, −6.68207245460146034213515198336, −6.08098847275896407389133336575, −5.12690324195238066573339516480, −4.93114980408101210325267720506, −3.47270454043981370096301100649, −3.21829294345997772812634167558, −1.96699586581646016048602001970, −0.816459198591755542034641427459,
0.816459198591755542034641427459, 1.96699586581646016048602001970, 3.21829294345997772812634167558, 3.47270454043981370096301100649, 4.93114980408101210325267720506, 5.12690324195238066573339516480, 6.08098847275896407389133336575, 6.68207245460146034213515198336, 7.11961595923255558568530422163, 8.130850228679215748899614826416
