L(s) = 1 | + (0.755 + 0.436i)2-s + (2.59 − 1.5i)3-s + (−3.61 − 6.26i)4-s + (10.8 + 2.81i)5-s + 2.61·6-s + (−10.7 − 15.0i)7-s − 13.2i·8-s + (4.5 − 7.79i)9-s + (6.94 + 6.84i)10-s + (10.5 + 18.2i)11-s + (−18.8 − 10.8i)12-s − 62.2i·13-s + (−1.56 − 16.0i)14-s + (32.3 − 8.92i)15-s + (−23.1 + 40.1i)16-s + (35.9 − 20.7i)17-s + ⋯ |
L(s) = 1 | + (0.267 + 0.154i)2-s + (0.499 − 0.288i)3-s + (−0.452 − 0.783i)4-s + (0.967 + 0.251i)5-s + 0.178·6-s + (−0.581 − 0.813i)7-s − 0.587i·8-s + (0.166 − 0.288i)9-s + (0.219 + 0.216i)10-s + (0.288 + 0.500i)11-s + (−0.452 − 0.261i)12-s − 1.32i·13-s + (−0.0298 − 0.306i)14-s + (0.556 − 0.153i)15-s + (−0.361 + 0.626i)16-s + (0.512 − 0.295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.73175 - 1.11121i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73175 - 1.11121i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.59 + 1.5i)T \) |
| 5 | \( 1 + (-10.8 - 2.81i)T \) |
| 7 | \( 1 + (10.7 + 15.0i)T \) |
good | 2 | \( 1 + (-0.755 - 0.436i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-10.5 - 18.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 62.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-35.9 + 20.7i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-11.2 + 19.4i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-33.8 - 19.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 59.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-152. - 264. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (138. + 79.8i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 497.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 257. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-446. - 257. i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (36.1 - 20.8i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-136. - 235. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (371. - 642. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-661. + 381. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 42.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + (104. - 60.3i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-345. + 598. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.23e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-412. + 715. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 259. iT - 9.12e5T^{2} \) |
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
−13.41065984437210619835657972280, −12.49295007341106056043271839359, −10.45397748550955070996763834235, −10.02721479944390744899006070594, −8.933234088354384300685255871772, −7.26033512633038255527431419729, −6.24395485794589823851344800947, −4.96761772259004185433886097588, −3.18104676965702505771381936863, −1.13582536325352695988139226549,
2.31197398501390547633393623651, 3.70742641139745606012809148437, 5.18910407485920504058515858334, 6.57462654852069286518513837644, 8.380063592171532298602582303003, 9.092431305654168173237971982465, 9.953831608573851063253039924867, 11.66820789821309956129346306130, 12.53543054385781004626179233655, 13.62550628053777075053622346169