L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.30 + 1.30i)3-s + 1.00i·4-s + (0.158 + 2.23i)5-s − 1.84i·6-s + (−0.941 − 2.47i)7-s + (0.707 − 0.707i)8-s + 0.414i·9-s + (1.46 − 1.68i)10-s + 2.82·11-s + (−1.30 + 1.30i)12-s + (−4.23 − 4.23i)13-s + (−1.08 + 2.41i)14-s + (−2.70 + 3.12i)15-s − 1.00·16-s + (−3.69 + 3.69i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.754 + 0.754i)3-s + 0.500i·4-s + (0.0708 + 0.997i)5-s − 0.754i·6-s + (−0.355 − 0.934i)7-s + (0.250 − 0.250i)8-s + 0.138i·9-s + (0.463 − 0.534i)10-s + 0.852·11-s + (−0.377 + 0.377i)12-s + (−1.17 − 1.17i)13-s + (−0.289 + 0.645i)14-s + (−0.698 + 0.805i)15-s − 0.250·16-s + (−0.896 + 0.896i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.866432 + 0.0882676i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.866432 + 0.0882676i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.158 - 2.23i)T \) |
| 7 | \( 1 + (0.941 + 2.47i)T \) |
good | 3 | \( 1 + (-1.30 - 1.30i)T + 3iT^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + (4.23 + 4.23i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.69 - 3.69i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.39T + 19T^{2} \) |
| 23 | \( 1 + (-0.414 + 0.414i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.828iT - 29T^{2} \) |
| 31 | \( 1 - 1.53iT - 31T^{2} \) |
| 37 | \( 1 + (-2.58 - 2.58i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.69iT - 41T^{2} \) |
| 43 | \( 1 + (-4 + 4i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.08 + 1.08i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.24 - 8.24i)T - 53iT^{2} \) |
| 59 | \( 1 + 9.23T + 59T^{2} \) |
| 61 | \( 1 + 6.43iT - 61T^{2} \) |
| 67 | \( 1 + (-10.4 - 10.4i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.585T + 71T^{2} \) |
| 73 | \( 1 + (-4.14 - 4.14i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.07iT - 79T^{2} \) |
| 83 | \( 1 + (-5.31 - 5.31i)T + 83iT^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + (4.59 - 4.59i)T - 97iT^{2} \) |
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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
−14.78470987378120487362941040263, −13.85977008777389613366498635225, −12.50552335021490797127893041958, −10.99134125831819050218991468451, −10.14868686409161879514923571805, −9.409940639487214685885581009183, −7.921038775700212950159562894875, −6.65806377191328104629730671593, −4.07963943454371272144407811349, −2.93893812303338789770631387506,
2.11189885120549725243992584967, 4.88199365542020945463572457055, 6.56757197148260786786250918424, 7.75192276755752972122424539104, 9.124475990955019392277946217372, 9.303791439939596796170436166651, 11.63020594915862570810532593404, 12.58552378802120746593658974618, 13.73460963481235310312476185764, 14.59657449981060710725166800223