| L(s) = 1 | + (0.119 − 1.13i)2-s + (−0.709 − 1.57i)3-s + (0.673 + 0.143i)4-s + 1.75·5-s + (−1.88 + 0.619i)6-s + (−0.444 − 1.36i)7-s + (0.951 − 2.92i)8-s + (−1.99 + 2.24i)9-s + (0.209 − 1.99i)10-s + (4.15 + 0.883i)11-s + (−0.251 − 1.16i)12-s + (−0.566 + 0.411i)13-s + (−1.61 + 0.342i)14-s + (−1.24 − 2.77i)15-s + (−1.96 − 0.874i)16-s + (−3.90 − 4.34i)17-s + ⋯ |
| L(s) = 1 | + (0.0846 − 0.805i)2-s + (−0.409 − 0.912i)3-s + (0.336 + 0.0715i)4-s + 0.784·5-s + (−0.769 + 0.252i)6-s + (−0.167 − 0.516i)7-s + (0.336 − 1.03i)8-s + (−0.664 + 0.747i)9-s + (0.0663 − 0.631i)10-s + (1.25 + 0.266i)11-s + (−0.0726 − 0.336i)12-s + (−0.157 + 0.114i)13-s + (−0.430 + 0.0915i)14-s + (−0.321 − 0.715i)15-s + (−0.491 − 0.218i)16-s + (−0.948 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.780887 - 1.27081i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.780887 - 1.27081i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.709 + 1.57i)T \) |
| 31 | \( 1 + (5.56 + 0.116i)T \) |
| good | 2 | \( 1 + (-0.119 + 1.13i)T + (-1.95 - 0.415i)T^{2} \) |
| 5 | \( 1 - 1.75T + 5T^{2} \) |
| 7 | \( 1 + (0.444 + 1.36i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (-4.15 - 0.883i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.566 - 0.411i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.90 + 4.34i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (4.54 - 2.02i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-5.95 - 6.61i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.0918 - 0.874i)T + (-28.3 - 6.02i)T^{2} \) |
| 37 | \( 1 + (1.21 - 2.10i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.95 + 3.60i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.67 - 3.39i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-7.45 - 3.31i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-5.71 + 1.21i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-10.6 - 4.74i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-2.86 - 4.97i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 - 4.03T + 67T^{2} \) |
| 71 | \( 1 + (1.95 - 0.415i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (4.11 - 4.57i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (2.78 - 8.55i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (6.30 - 2.80i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-1.77 - 5.46i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-11.1 + 12.4i)T + (-10.1 - 96.4i)T^{2} \) |
| 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
−11.55339339740376645436772908435, −10.93495931652189190969983810284, −9.858559617091145537280287390525, −8.916073997833372520782237248730, −7.13757495956930706235693826087, −6.88155258035733298525429814229, −5.63472303081581355603224666353, −4.00511819828763843771007931601, −2.41538694784634766283517179922, −1.34426661649453089458296157404,
2.30526427739738280216257705707, 4.12136841827857377042151020711, 5.36281876301310492955625704522, 6.23515133255741810437038625429, 6.76672093818643740369127852351, 8.662912918096384658268329714282, 9.063136259037388500989210604206, 10.43612615001851072488044011361, 11.02174397958240661322769909337, 12.03432149128685911648949012225
