| L(s) = 1 | − 0.792·2-s + (1.26 + 1.18i)3-s − 1.37·4-s + (−1 − 0.939i)6-s + (−1.73 − 2i)7-s + 2.67·8-s + (0.186 + 2.99i)9-s + 2.52i·11-s + (−1.73 − 1.62i)12-s − 4.10·13-s + (1.37 + 1.58i)14-s + 0.627·16-s + 4.37i·17-s + (−0.147 − 2.37i)18-s + 3.46i·19-s + ⋯ |
| L(s) = 1 | − 0.560·2-s + (0.728 + 0.684i)3-s − 0.686·4-s + (−0.408 − 0.383i)6-s + (−0.654 − 0.755i)7-s + 0.944·8-s + (0.0620 + 0.998i)9-s + 0.761i·11-s + (−0.499 − 0.469i)12-s − 1.13·13-s + (0.366 + 0.423i)14-s + 0.156·16-s + 1.06i·17-s + (−0.0347 − 0.559i)18-s + 0.794i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.137465 + 0.533592i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.137465 + 0.533592i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.26 - 1.18i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.73 + 2i)T \) |
| good | 2 | \( 1 + 0.792T + 2T^{2} \) |
| 11 | \( 1 - 2.52iT - 11T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 - 4.37iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 8.51T + 23T^{2} \) |
| 29 | \( 1 - 0.939iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 6.74iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4.74iT - 43T^{2} \) |
| 47 | \( 1 + 1.62iT - 47T^{2} \) |
| 53 | \( 1 - 1.87T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 4.74iT - 67T^{2} \) |
| 71 | \( 1 + 0.294iT - 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 2.37T + 79T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−10.60365852487482507659249872207, −10.04263210782151776330139821347, −9.721495671917088997751307741479, −8.631490537612222691955060108073, −7.86832227324924079384201629738, −7.06907283876123528535952795705, −5.46372559401622217586319613712, −4.28091842290024955264614465966, −3.72306542630018199758655764664, −1.99685524035399596224557046793,
0.34852288831556581176598045684, 2.24759570326835229254030108210, 3.35021021375821708529831650545, 4.76895803325990689116006038205, 6.00678346897378948177245964918, 7.08980069822029025876613823933, 8.019988862828137118644719273835, 8.686211865142630404464192079445, 9.590568846716256617361075319038, 9.912895478773522222873831730945
