L(s) = 1 | + 3.99i·2-s − 7.98·4-s + (−3.61 + 10.5i)5-s + 7.79i·7-s + 0.0620i·8-s + (−42.3 − 14.4i)10-s + 32.7·11-s + 65.9i·13-s − 31.1·14-s − 64.1·16-s + 15.5i·17-s − 51.6·19-s + (28.8 − 84.4i)20-s + 130. i·22-s + 49.6i·23-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 0.998·4-s + (−0.323 + 0.946i)5-s + 0.421i·7-s + 0.00274i·8-s + (−1.33 − 0.456i)10-s + 0.896·11-s + 1.40i·13-s − 0.595·14-s − 1.00·16-s + 0.222i·17-s − 0.623·19-s + (0.322 − 0.944i)20-s + 1.26i·22-s + 0.450i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.311842276\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.311842276\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (3.61 - 10.5i)T \) |
good | 2 | \( 1 - 3.99iT - 8T^{2} \) |
| 7 | \( 1 - 7.79iT - 343T^{2} \) |
| 11 | \( 1 - 32.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 15.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 51.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 282.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 122. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 109.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 191. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 414. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 236. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 784.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 535.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 121. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 590.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 986. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 607.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 610. iT - 9.12e5T^{2} \) |
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
−11.55907724581170739790351026250, −10.53556680428266796988655226296, −9.262675287838411916095414418396, −8.570074064442885733686779809812, −7.52170721405154637893201290082, −6.59590980292324639356247872692, −6.29991282456373578716902377741, −4.86917246501143723300525642587, −3.75175313167767002382161146576, −2.11221747861135453171227275832,
0.46019798840831088073270643596, 1.37428827520788215461600059400, 2.90567815590252675384135628049, 3.99906930749119362732476346892, 4.82312726763114911224464773780, 6.27580577195082117374025687570, 7.63454449034131721839255562698, 8.676359808014508913423175844076, 9.457051466850229004130440804064, 10.43363385519694239933257580834