| L(s) = 1 | − 1.41·2-s + (−2.16 − 2.07i)3-s + 2.00·4-s + (4.21 − 2.69i)5-s + (3.06 + 2.93i)6-s − 3.87i·7-s − 2.82·8-s + (0.399 + 8.99i)9-s + (−5.95 + 3.80i)10-s + 5.04i·11-s + (−4.33 − 4.14i)12-s − 15.3i·13-s + 5.47i·14-s + (−14.7 − 2.89i)15-s + 4.00·16-s − 10.2·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.722 − 0.691i)3-s + 0.500·4-s + (0.842 − 0.538i)5-s + (0.510 + 0.488i)6-s − 0.553i·7-s − 0.353·8-s + (0.0444 + 0.999i)9-s + (−0.595 + 0.380i)10-s + 0.458i·11-s + (−0.361 − 0.345i)12-s − 1.18i·13-s + 0.391i·14-s + (−0.981 − 0.193i)15-s + 0.250·16-s − 0.602·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5195648171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5195648171\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (2.16 + 2.07i)T \) |
| 5 | \( 1 + (-4.21 + 2.69i)T \) |
| 23 | \( 1 - 4.79T \) |
| good | 7 | \( 1 + 3.87iT - 49T^{2} \) |
| 11 | \( 1 - 5.04iT - 121T^{2} \) |
| 13 | \( 1 + 15.3iT - 169T^{2} \) |
| 17 | \( 1 + 10.2T + 289T^{2} \) |
| 19 | \( 1 + 19.1T + 361T^{2} \) |
| 29 | \( 1 + 10.8iT - 841T^{2} \) |
| 31 | \( 1 - 0.921T + 961T^{2} \) |
| 37 | \( 1 + 31.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 28.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 26.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 53.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 15.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 23.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 55.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 43.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 2.55iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 12.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 101.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 146.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 126. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 136. iT - 9.40e3T^{2} \) |
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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
−10.02661669317383303757798646170, −8.886244500342217548567098446238, −8.057720206578160797865481375999, −7.17283329512882114882326527264, −6.31985976483845417222765721407, −5.52054980168674037252330636888, −4.45669306557087119131285724839, −2.52369883476892682348402961795, −1.43897812579274828273007758350, −0.25551126152523544298097012995,
1.69488978182746023968402384053, 2.95691441497707736187659827081, 4.37019296897005933918120296257, 5.53515248922731058641183608549, 6.39585960474934065570295839998, 6.88329885761589967246403948108, 8.496623660436765683020846829071, 9.180536583436860842391920989885, 9.816430446082965866517307816449, 10.73757121419315913564877301896
