L(s) = 1 | + 2-s + (1 + 1.41i)3-s + 4-s − 1.41i·5-s + (1 + 1.41i)6-s − 4·7-s + 8-s + (−1.00 + 2.82i)9-s − 1.41i·10-s − 5.65i·11-s + (1 + 1.41i)12-s + 4.24i·13-s − 4·14-s + (2.00 − 1.41i)15-s + 16-s + 2.82i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.577 + 0.816i)3-s + 0.5·4-s − 0.632i·5-s + (0.408 + 0.577i)6-s − 1.51·7-s + 0.353·8-s + (−0.333 + 0.942i)9-s − 0.447i·10-s − 1.70i·11-s + (0.288 + 0.408i)12-s + 1.17i·13-s − 1.06·14-s + (0.516 − 0.365i)15-s + 0.250·16-s + 0.685i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.54269 + 0.299894i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.54269 + 0.299894i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 - 1.41i)T \) |
| 19 | \( 1 + (-1 + 4.24i)T \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 1.41iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 4.24iT - 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8.48iT - 97T^{2} \) |
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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
−13.53467691744317623608872832408, −13.02674724024610970578432370356, −11.60928323620627003007230293400, −10.54303899595967610980929629665, −9.275813778740862873684277615043, −8.587510014585810766821465324872, −6.73832795362808146064718524367, −5.51443916036930876837155240244, −4.04605938820785517167919607874, −3.00489611670191032575990616124,
2.52754969442324543840005105817, 3.64773973712878028607102946935, 5.77270130908789855328915338266, 6.94909488668822592649725569771, 7.56995308509356553363623823211, 9.433711982995668883810367783194, 10.28203756050024740552854529559, 11.93527617100321057580621015372, 12.78992616265785395876288403106, 13.25092394538342122552329217174