L(s) = 1 | + (−4.19 + 2.41i)3-s + (0.0446 + 0.0257i)5-s + (6.12 + 3.39i)7-s + (7.20 − 12.4i)9-s + (−0.894 − 1.55i)11-s − 5.87i·13-s − 0.249·15-s + (23.0 − 13.2i)17-s + (22.8 + 13.1i)19-s + (−33.8 + 0.603i)21-s + (−12.8 + 22.2i)23-s + (−12.4 − 21.6i)25-s + 26.1i·27-s + 27.1·29-s + (−25.7 + 14.8i)31-s + ⋯ |
L(s) = 1 | + (−1.39 + 0.806i)3-s + (0.00892 + 0.00515i)5-s + (0.874 + 0.484i)7-s + (0.800 − 1.38i)9-s + (−0.0813 − 0.140i)11-s − 0.452i·13-s − 0.0166·15-s + (1.35 − 0.781i)17-s + (1.20 + 0.693i)19-s + (−1.61 + 0.0287i)21-s + (−0.558 + 0.966i)23-s + (−0.499 − 0.865i)25-s + 0.969i·27-s + 0.937·29-s + (−0.829 + 0.479i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.120465894\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.120465894\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-6.12 - 3.39i)T \) |
good | 3 | \( 1 + (4.19 - 2.41i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-0.0446 - 0.0257i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (0.894 + 1.55i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 5.87iT - 169T^{2} \) |
| 17 | \( 1 + (-23.0 + 13.2i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-22.8 - 13.1i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (12.8 - 22.2i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 27.1T + 841T^{2} \) |
| 31 | \( 1 + (25.7 - 14.8i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (30.8 - 53.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 65.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 9.52T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-61.2 - 35.3i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-4.86 - 8.42i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-54.3 + 31.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-66.1 - 38.2i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (51.5 + 89.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 90.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + (28.8 - 16.6i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (32.4 - 56.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 29.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-18.7 - 10.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 123. iT - 9.40e3T^{2} \) |
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
−11.21018738189768287242254397122, −10.14217889071938775009748928674, −9.759558748682660804364822598963, −8.334966344836926697134490328795, −7.41277249874377575680822812652, −5.94260943775371375446453245179, −5.42338354983395840988445178483, −4.63003093306972320449784048914, −3.24833437193241313517256754695, −1.15331925877848861674724393644,
0.70249484965947094230098591327, 1.86440027122723138025885839038, 3.95454389591663004065372333102, 5.24497349184444841964016492403, 5.78293445560339113842735709795, 7.13306312753678541949461313567, 7.48148173186864906385413205356, 8.743388787327962621170492762474, 10.18064216351976458397605912115, 10.78983120028155384228432867731