L(s) = 1 | + (0.956 − 1.65i)2-s + (−0.830 − 1.43i)4-s + (1.54 + 1.61i)5-s + (1.11 + 2.39i)7-s + 0.650·8-s + (4.15 − 1.00i)10-s + (2.79 − 1.61i)11-s − 4.86·13-s + (5.04 + 0.439i)14-s + (2.28 − 3.95i)16-s + (0.631 − 0.364i)17-s + (−6.81 − 3.93i)19-s + (1.04 − 3.56i)20-s − 6.17i·22-s + (−2.43 + 4.21i)23-s + ⋯ |
L(s) = 1 | + (0.676 − 1.17i)2-s + (−0.415 − 0.718i)4-s + (0.689 + 0.723i)5-s + (0.422 + 0.906i)7-s + 0.229·8-s + (1.31 − 0.318i)10-s + (0.842 − 0.486i)11-s − 1.34·13-s + (1.34 + 0.117i)14-s + (0.570 − 0.988i)16-s + (0.153 − 0.0884i)17-s + (−1.56 − 0.902i)19-s + (0.234 − 0.796i)20-s − 1.31i·22-s + (−0.507 + 0.879i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.89104 - 0.950324i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.89104 - 0.950324i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.54 - 1.61i)T \) |
| 7 | \( 1 + (-1.11 - 2.39i)T \) |
good | 2 | \( 1 + (-0.956 + 1.65i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.79 + 1.61i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.86T + 13T^{2} \) |
| 17 | \( 1 + (-0.631 + 0.364i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.81 + 3.93i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.43 - 4.21i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.75iT - 29T^{2} \) |
| 31 | \( 1 + (1.23 - 0.714i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.74 - 1.58i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.40T + 41T^{2} \) |
| 43 | \( 1 + 6.42iT - 43T^{2} \) |
| 47 | \( 1 + (-4.21 - 2.43i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.760 - 1.31i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.15 + 5.45i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.05 + 1.18i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.63 - 5.56i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 + (-6.91 - 11.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.99 - 3.44i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.19iT - 83T^{2} \) |
| 89 | \( 1 + (5.63 - 9.76i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
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
−11.59688067715526654060126260494, −10.86402536434585512584489684071, −9.886675958918217698448606835712, −9.087905703892293257264089873818, −7.66072775110806690656107412122, −6.38887814136873039469652245769, −5.32344004037930374965915743459, −4.15599667919447131101257092936, −2.74098229517502246319073965870, −2.01204395260449452813461620913,
1.75946505666453335305867922731, 4.21838679833167901726571093157, 4.74099561977296451506902605084, 5.93488023537242107289207410978, 6.81955060980517725143726351471, 7.72016933548646511957698817792, 8.722120319868496593215921571457, 9.943864180972521185029268518729, 10.67770750334351795183645471122, 12.29790007883317118719504183748