L(s) = 1 | + (−1.91 − 1.91i)2-s + (−0.520 + 1.65i)3-s + 5.34i·4-s + (0.478 + 2.18i)5-s + (4.16 − 2.16i)6-s + (−1.54 + 1.54i)7-s + (6.41 − 6.41i)8-s + (−2.45 − 1.72i)9-s + (3.26 − 5.10i)10-s + 2.77i·11-s + (−8.82 − 2.78i)12-s + (−3.18 − 3.18i)13-s + 5.93·14-s + (−3.85 − 0.347i)15-s − 13.8·16-s + (−3.06 − 3.06i)17-s + ⋯ |
L(s) = 1 | + (−1.35 − 1.35i)2-s + (−0.300 + 0.953i)3-s + 2.67i·4-s + (0.213 + 0.976i)5-s + (1.69 − 0.885i)6-s + (−0.585 + 0.585i)7-s + (2.26 − 2.26i)8-s + (−0.819 − 0.573i)9-s + (1.03 − 1.61i)10-s + 0.836i·11-s + (−2.54 − 0.803i)12-s + (−0.884 − 0.884i)13-s + 1.58·14-s + (−0.995 − 0.0897i)15-s − 3.46·16-s + (−0.744 − 0.744i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0627419 + 0.188242i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0627419 + 0.188242i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.520 - 1.65i)T \) |
| 5 | \( 1 + (-0.478 - 2.18i)T \) |
| 19 | \( 1 + iT \) |
good | 2 | \( 1 + (1.91 + 1.91i)T + 2iT^{2} \) |
| 7 | \( 1 + (1.54 - 1.54i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.77iT - 11T^{2} \) |
| 13 | \( 1 + (3.18 + 3.18i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.06 + 3.06i)T + 17iT^{2} \) |
| 23 | \( 1 + (-1.05 + 1.05i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.341T + 29T^{2} \) |
| 31 | \( 1 - 6.05T + 31T^{2} \) |
| 37 | \( 1 + (5.33 - 5.33i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.460iT - 41T^{2} \) |
| 43 | \( 1 + (3.43 + 3.43i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.93 - 1.93i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.19 - 4.19i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + (2.96 - 2.96i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.44iT - 71T^{2} \) |
| 73 | \( 1 + (-11.0 - 11.0i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.95iT - 79T^{2} \) |
| 83 | \( 1 + (1.98 - 1.98i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.63T + 89T^{2} \) |
| 97 | \( 1 + (7.00 - 7.00i)T - 97iT^{2} \) |
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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
−11.89680597336349802854607423577, −10.92300821188777658186897160627, −10.25520694628231502692472103694, −9.664187518067004176276231402137, −8.983381647884655683091878494624, −7.68157576309244000921879509123, −6.55646630478705924973955584919, −4.71281839504139866270363002085, −3.16766343465129340006987986090, −2.53232474817516235309881234443,
0.23187970526397696969078672502, 1.68080470178526485380632049063, 4.82010999960552267092009311247, 5.98621100945657427225498080635, 6.64072542753284542100562927281, 7.58974149765807765209154515940, 8.472068241314308838159923180009, 9.171245019500982337350717888244, 10.17572229972920905597967044498, 11.21094185248778653334123755218