| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 2.32·7-s + 8-s + 9-s − 10-s + 5.34·11-s + 12-s + 2.32·14-s − 15-s + 16-s + 4·17-s + 18-s − 4.02·19-s − 20-s + 2.32·21-s + 5.34·22-s − 4.93·23-s + 24-s + 25-s + 27-s + 2.32·28-s + 4.29·29-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 + 0.877·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.61·11-s + 0.288·12-s + 0.620·14-s − 0.258·15-s + 0.250·16-s + 0.970·17-s + 0.235·18-s − 0.922·19-s − 0.223·20-s + 0.506·21-s + 1.13·22-s − 1.02·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.438·28-s + 0.796·29-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
| \(L(1)\) |
\(\approx\) |
\(4.667308511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.667308511\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.32T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 23 | \( 1 + 4.93T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 - 3.47T + 31T^{2} \) |
| 37 | \( 1 + 3.14T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 1.81T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 61 | \( 1 - 0.535T + 61T^{2} \) |
| 67 | \( 1 + 4.10T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 1.73T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.133791637941514548912906329855, −7.60405675959653604640735072839, −6.68914949466887446603248153959, −6.15814682468705949321951861614, −5.15044231209152981611987545800, −4.25951504614862268231986319258, −3.95940720228490488937765495374, −3.01078960066989012448692542507, −1.96720042109478384111837228888, −1.13917446798670816549716940848,
1.13917446798670816549716940848, 1.96720042109478384111837228888, 3.01078960066989012448692542507, 3.95940720228490488937765495374, 4.25951504614862268231986319258, 5.15044231209152981611987545800, 6.15814682468705949321951861614, 6.68914949466887446603248153959, 7.60405675959653604640735072839, 8.133791637941514548912906329855
