L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.66 − 0.492i)3-s − 1.00i·4-s + (1.86 − 1.23i)5-s + (−1.52 + 0.825i)6-s + (−3.33 − 3.33i)7-s + (−0.707 − 0.707i)8-s + (2.51 + 1.63i)9-s + (0.442 − 2.19i)10-s + 0.00329i·11-s + (−0.492 + 1.66i)12-s + (1.14 − 1.14i)13-s − 4.71·14-s + (−3.70 + 1.13i)15-s − 1.00·16-s + (−4.95 + 4.95i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.958 − 0.284i)3-s − 0.500i·4-s + (0.833 − 0.553i)5-s + (−0.621 + 0.337i)6-s + (−1.26 − 1.26i)7-s + (−0.250 − 0.250i)8-s + (0.838 + 0.545i)9-s + (0.139 − 0.693i)10-s + 0.000994i·11-s + (−0.142 + 0.479i)12-s + (0.318 − 0.318i)13-s − 1.26·14-s + (−0.956 + 0.293i)15-s − 0.250·16-s + (−1.20 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0335011 - 1.01726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0335011 - 1.01726i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.66 + 0.492i)T \) |
| 5 | \( 1 + (-1.86 + 1.23i)T \) |
| 19 | \( 1 + iT \) |
good | 7 | \( 1 + (3.33 + 3.33i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.00329iT - 11T^{2} \) |
| 13 | \( 1 + (-1.14 + 1.14i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.95 - 4.95i)T - 17iT^{2} \) |
| 23 | \( 1 + (1.28 + 1.28i)T + 23iT^{2} \) |
| 29 | \( 1 - 1.91T + 29T^{2} \) |
| 31 | \( 1 + 3.49T + 31T^{2} \) |
| 37 | \( 1 + (6.71 + 6.71i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.63iT - 41T^{2} \) |
| 43 | \( 1 + (-1.52 + 1.52i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.07 - 2.07i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.97 - 5.97i)T + 53iT^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 1.95T + 61T^{2} \) |
| 67 | \( 1 + (5.56 + 5.56i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.22iT - 71T^{2} \) |
| 73 | \( 1 + (-10.3 + 10.3i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.5iT - 79T^{2} \) |
| 83 | \( 1 + (-9.29 - 9.29i)T + 83iT^{2} \) |
| 89 | \( 1 + 6.59T + 89T^{2} \) |
| 97 | \( 1 + (-7.72 - 7.72i)T + 97iT^{2} \) |
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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
−10.56928027077264478797297984320, −9.808703549420903566020030490878, −8.780766487984251923689724478838, −7.21762378144715786164939251224, −6.40372105572771665377625750761, −5.79595841555664150560016663174, −4.59488955864292812701395818632, −3.71087422000719690708267010572, −1.96059768695099538659483878162, −0.53003742285023130351779906069,
2.37921211161760915801928554599, 3.55209476834965610605940568272, 5.02314702478334885501130812390, 5.73538665032153681850986374258, 6.56091003485156808364791481668, 6.92629422837700065854218592419, 8.755682789978961292206895329892, 9.489626495842438115141704266306, 10.18885735805102574728093139741, 11.38846719243295943162017945682