L(s) = 1 | + (0.190 − 0.109i)2-s + (0.800 + 1.38i)3-s + (−0.975 + 1.69i)4-s + i·5-s + (0.304 + 0.175i)6-s + (0.287 + 0.166i)7-s + 0.868i·8-s + (0.219 − 0.380i)9-s + (0.109 + 0.190i)10-s + (−4.65 + 2.68i)11-s − 3.12·12-s + 0.0729·14-s + (−1.38 + 0.800i)15-s + (−1.85 − 3.21i)16-s + (−2.53 + 4.38i)17-s − 0.0965i·18-s + ⋯ |
L(s) = 1 | + (0.134 − 0.0776i)2-s + (0.461 + 0.800i)3-s + (−0.487 + 0.845i)4-s + 0.447i·5-s + (0.124 + 0.0717i)6-s + (0.108 + 0.0627i)7-s + 0.306i·8-s + (0.0732 − 0.126i)9-s + (0.0347 + 0.0601i)10-s + (−1.40 + 0.809i)11-s − 0.901·12-s + 0.0195·14-s + (−0.357 + 0.206i)15-s + (−0.464 − 0.803i)16-s + (−0.614 + 1.06i)17-s − 0.0227i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.155662 + 1.21240i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.155662 + 1.21240i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.190 + 0.109i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.800 - 1.38i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.287 - 0.166i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.65 - 2.68i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.53 - 4.38i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.96 - 1.13i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.41 + 2.45i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.45 - 2.51i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.46iT - 31T^{2} \) |
| 37 | \( 1 + (-5.17 + 2.98i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.23 - 1.86i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.53 - 4.38i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 8.34iT - 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 + (2.34 + 1.35i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.05 - 12.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.94 - 5.16i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.0 - 6.39i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 9.68iT - 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 - 4.26iT - 83T^{2} \) |
| 89 | \( 1 + (-2.79 + 1.61i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.17 + 1.25i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.38300342994373651388198129742, −9.797187532723001660385424081270, −8.912554135190813089886502246957, −8.073723849054669764539181598053, −7.45141166075267481544241799595, −6.22221239196319680486448813919, −4.88319631731217711452687797384, −4.24429743522534153261940345601, −3.28013464871661020532157837796, −2.36523694123015608002082040917,
0.52878220703345017852037466947, 1.88537350915967491844950530027, 3.09668163842090058156167352864, 4.73317746548204889142991667385, 5.22440407676893868737678496809, 6.30317461378459299053206207814, 7.33377526892525332381399046544, 8.129049681119132815115010374833, 8.841882285746706545056354320614, 9.764032500340305220204705520658