L(s) = 1 | + (0.127 − 0.393i)2-s + (−2.28 + 1.66i)3-s + (1.47 + 1.07i)4-s + (−0.309 − 0.951i)5-s + (0.362 + 1.11i)6-s + (−1.61 − 1.17i)7-s + (1.28 − 0.932i)8-s + (1.54 − 4.75i)9-s − 0.414·10-s − 5.17·12-s + (2.11 − 6.49i)13-s + (−0.670 + 0.486i)14-s + (2.28 + 1.66i)15-s + (0.927 + 2.85i)16-s + (−0.362 − 1.11i)17-s + (−1.67 − 1.21i)18-s + ⋯ |
L(s) = 1 | + (0.0905 − 0.278i)2-s + (−1.32 + 0.959i)3-s + (0.739 + 0.537i)4-s + (−0.138 − 0.425i)5-s + (0.147 + 0.454i)6-s + (−0.611 − 0.444i)7-s + (0.453 − 0.329i)8-s + (0.515 − 1.58i)9-s − 0.130·10-s − 1.49·12-s + (0.585 − 1.80i)13-s + (−0.179 + 0.130i)14-s + (0.590 + 0.429i)15-s + (0.231 + 0.713i)16-s + (−0.0878 − 0.270i)17-s + (−0.394 − 0.286i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02486 - 0.255832i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02486 - 0.255832i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.127 + 0.393i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (2.28 - 1.66i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (1.61 + 1.17i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-2.11 + 6.49i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.362 + 1.11i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + (-6.19 - 4.50i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.95 + 2.14i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.85 + 3.52i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (-2.28 + 1.66i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.106 + 0.326i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.81 - 5.67i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (4.11 + 12.6i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 71 | \( 1 + (3.49 + 10.7i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (5.52 + 4.01i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (1.23 - 3.80i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.85 - 5.70i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 9.31T + 89T^{2} \) |
| 97 | \( 1 + (2.36 - 7.28i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.58014824704398885086092655919, −10.25652977586491593236120269116, −9.020261380785152340016355010968, −7.85087151347450406111493244860, −6.82026141146853880411811550371, −5.91357303320321488455808145183, −5.04228174135960546839921953981, −3.92349245121681516457866480384, −3.06936642421916171185677208465, −0.75639729838863468353415584226,
1.30301107886333873325601333491, 2.53473080909625189039432969968, 4.43464313202521559223413210483, 5.69955885786080111162698146582, 6.36422436304158331162700240688, 6.75358635196571883653271200906, 7.59208254103792757710015168944, 8.976473233524230955904288697341, 10.15021625576368863822845465056, 10.99284793698350527813499150318