| L(s) = 1 | + 2-s − 1.23·3-s + 4-s + 1.23·5-s − 1.23·6-s + 8-s − 1.47·9-s + 1.23·10-s + 11-s − 1.23·12-s + 3.23·13-s − 1.52·15-s + 16-s − 2.47·17-s − 1.47·18-s + 7.23·19-s + 1.23·20-s + 22-s + 4·23-s − 1.23·24-s − 3.47·25-s + 3.23·26-s + 5.52·27-s + 4.47·29-s − 1.52·30-s − 2·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.713·3-s + 0.5·4-s + 0.552·5-s − 0.504·6-s + 0.353·8-s − 0.490·9-s + 0.390·10-s + 0.301·11-s − 0.356·12-s + 0.897·13-s − 0.394·15-s + 0.250·16-s − 0.599·17-s − 0.346·18-s + 1.66·19-s + 0.276·20-s + 0.213·22-s + 0.834·23-s − 0.252·24-s − 0.694·25-s + 0.634·26-s + 1.06·27-s + 0.830·29-s − 0.278·30-s − 0.359·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.243815542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.243815542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 - 0.763T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00829958563360915913838911374, −9.142548431920584778533938054409, −8.195523686917400799090646390594, −7.03886884663017761430529089934, −6.27285384438401664958786921833, −5.60433448207109467768543833834, −4.89768914171678265713289991347, −3.69541219660826411839179615129, −2.65054860399318456182246322213, −1.16174622122432376623073397048,
1.16174622122432376623073397048, 2.65054860399318456182246322213, 3.69541219660826411839179615129, 4.89768914171678265713289991347, 5.60433448207109467768543833834, 6.27285384438401664958786921833, 7.03886884663017761430529089934, 8.195523686917400799090646390594, 9.142548431920584778533938054409, 10.00829958563360915913838911374
