| L(s) = 1 | + 2.11i·3-s + i·5-s + 7-s − 1.48·9-s − 2.75i·11-s − 3.19i·13-s − 2.11·15-s + 0.352·17-s − 8.13i·19-s + 2.11i·21-s + 7.61·23-s − 25-s + 3.21i·27-s − 2.84i·29-s + 8.60·31-s + ⋯ |
| L(s) = 1 | + 1.22i·3-s + 0.447i·5-s + 0.377·7-s − 0.493·9-s − 0.831i·11-s − 0.886i·13-s − 0.546·15-s + 0.0856·17-s − 1.86i·19-s + 0.461i·21-s + 1.58·23-s − 0.200·25-s + 0.618i·27-s − 0.528i·29-s + 1.54·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.874648618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.874648618\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 2.11iT - 3T^{2} \) |
| 11 | \( 1 + 2.75iT - 11T^{2} \) |
| 13 | \( 1 + 3.19iT - 13T^{2} \) |
| 17 | \( 1 - 0.352T + 17T^{2} \) |
| 19 | \( 1 + 8.13iT - 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 + 2.84iT - 29T^{2} \) |
| 31 | \( 1 - 8.60T + 31T^{2} \) |
| 37 | \( 1 + 8.38iT - 37T^{2} \) |
| 41 | \( 1 + 6.89T + 41T^{2} \) |
| 43 | \( 1 + 10.8iT - 43T^{2} \) |
| 47 | \( 1 - 4.41T + 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 13.3iT - 59T^{2} \) |
| 61 | \( 1 - 4.55iT - 61T^{2} \) |
| 67 | \( 1 - 4.63iT - 67T^{2} \) |
| 71 | \( 1 + 1.00T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 4.60T + 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
| 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
−8.985379233262069382391544672809, −8.625710735008742068703801276784, −7.49253173937024965229279073867, −6.78975137634306540925387847894, −5.66595855518032443124924730752, −5.03854832678613909022147810456, −4.25944443443037198416563305061, −3.26746645564978018916292442580, −2.62416283532657441622181826144, −0.73627195086869726016001234796,
1.26303433505385949938591075420, 1.73173246940126987163870265566, 2.98298974257013264548919958127, 4.31264272744210950236831046926, 4.96692640999287727841028395306, 6.09390423069289340872354556787, 6.75556498215903084162739947085, 7.44316252372402801180448186527, 8.149189279814066802624544911082, 8.759486567301696115038294055637
