| L(s) = 1 | − 0.414i·2-s − 2.82i·3-s + 1.82·4-s − 1.17·6-s − 2i·7-s − 1.58i·8-s − 5.00·9-s + 11-s − 5.17i·12-s + 6.82i·13-s − 0.828·14-s + 3·16-s + 1.17i·17-s + 2.07i·18-s − 5.65·21-s − 0.414i·22-s + ⋯ |
| L(s) = 1 | − 0.292i·2-s − 1.63i·3-s + 0.914·4-s − 0.478·6-s − 0.755i·7-s − 0.560i·8-s − 1.66·9-s + 0.301·11-s − 1.49i·12-s + 1.89i·13-s − 0.221·14-s + 0.750·16-s + 0.284i·17-s + 0.488i·18-s − 1.23·21-s − 0.0883i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.797967 - 1.29113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.797967 - 1.29113i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.414iT - 2T^{2} \) |
| 3 | \( 1 + 2.82iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 - 6.82iT - 13T^{2} \) |
| 17 | \( 1 - 1.17iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3.65iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 0.343iT - 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4.48iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 6.82iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 9.31T + 89T^{2} \) |
| 97 | \( 1 + 7.65iT - 97T^{2} \) |
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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
−11.59295244178732043903599753799, −11.12360429289197452009420931951, −9.760389359797121367145887006245, −8.454979523558487280598901720878, −7.26990349252116302854572442786, −6.89395966359866780253505746893, −6.02259013673371578289011117629, −3.99854168546424015710266382145, −2.32687646222128897370739740071, −1.33719664321169824589643543215,
2.67464430784910526810956774364, 3.73815666774460920540588790787, 5.40206477674209174978974510786, 5.72171463244198737573255066967, 7.39265919229779502224473295362, 8.478838844715940628802910640840, 9.464258081518195992711099351224, 10.37280365643641391671080200478, 11.05769676662173517101600625210, 11.89764355349030160910439024529
