L(s) = 1 | + (0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 − 0.499i)4-s + (−1.99 − 1.01i)5-s + (0.499 + 0.866i)6-s + (−0.219 + 0.219i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (−2.18 − 0.465i)10-s + 6.33·11-s + (0.707 + 0.707i)12-s + (3.59 + 0.962i)13-s + (−0.155 + 0.268i)14-s + (0.465 − 2.18i)15-s + (0.500 − 0.866i)16-s + (0.885 + 3.30i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + (−0.890 − 0.454i)5-s + (0.204 + 0.353i)6-s + (−0.0828 + 0.0828i)7-s + (0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.691 − 0.147i)10-s + 1.90·11-s + (0.204 + 0.204i)12-s + (0.996 + 0.267i)13-s + (−0.0414 + 0.0717i)14-s + (0.120 − 0.564i)15-s + (0.125 − 0.216i)16-s + (0.214 + 0.801i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.21262 + 0.0237916i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21262 + 0.0237916i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 + (1.99 + 1.01i)T \) |
| 19 | \( 1 + (-3.97 + 1.79i)T \) |
good | 7 | \( 1 + (0.219 - 0.219i)T - 7iT^{2} \) |
| 11 | \( 1 - 6.33T + 11T^{2} \) |
| 13 | \( 1 + (-3.59 - 0.962i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (-0.885 - 3.30i)T + (-14.7 + 8.5i)T^{2} \) |
| 23 | \( 1 + (-0.601 + 2.24i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (3.42 + 5.93i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.71iT - 31T^{2} \) |
| 37 | \( 1 + (-0.817 - 0.817i)T + 37iT^{2} \) |
| 41 | \( 1 + (8.81 + 5.08i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (8.44 - 2.26i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (10.1 + 2.73i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.96 - 1.06i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.367 + 0.636i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.17 - 3.76i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.33 - 4.98i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (1.14 + 0.663i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.65 + 1.24i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.21 + 12.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.190 + 0.190i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.18 - 8.97i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (15.9 - 4.26i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.05932787723645112520826529536, −9.862726958058077351010228148813, −8.969832350996579293132784732308, −8.270058412482333830381923052055, −6.96013180956637083567624094106, −6.09776482671504660401380702241, −4.87225841808326873245531365409, −3.91612016688123667737156674441, −3.43842845463303012714149577929, −1.41162700999618990440542151772,
1.39540547128346122896817246017, 3.32961324162913203876089498862, 3.72590157040067348262454303205, 5.17095262211313942714408155259, 6.45500086714140129043792864879, 6.92752442073791454949781812775, 7.87803958567129329610836419364, 8.770700307079163507536572363513, 9.846983250117558172657455053154, 11.31510486565505405617873878328