L(s) = 1 | + (0.707 + 0.707i)2-s + (−1.56 − 0.731i)3-s + 1.00i·4-s + (1.45 + 1.69i)5-s + (−0.592 − 1.62i)6-s + (1.76 − 1.76i)7-s + (−0.707 + 0.707i)8-s + (1.92 + 2.29i)9-s + (−0.171 + 2.22i)10-s − 5.31i·11-s + (0.731 − 1.56i)12-s + (3.35 + 3.35i)13-s + 2.49·14-s + (−1.04 − 3.73i)15-s − 1.00·16-s + (−2.77 − 2.77i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.906 − 0.422i)3-s + 0.500i·4-s + (0.650 + 0.759i)5-s + (−0.241 − 0.664i)6-s + (0.666 − 0.666i)7-s + (−0.250 + 0.250i)8-s + (0.642 + 0.765i)9-s + (−0.0542 + 0.705i)10-s − 1.60i·11-s + (0.211 − 0.453i)12-s + (0.931 + 0.931i)13-s + 0.666·14-s + (−0.269 − 0.963i)15-s − 0.250·16-s + (−0.672 − 0.672i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86749 + 0.481033i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86749 + 0.481033i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.56 + 0.731i)T \) |
| 5 | \( 1 + (-1.45 - 1.69i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + (-1.76 + 1.76i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.31iT - 11T^{2} \) |
| 13 | \( 1 + (-3.35 - 3.35i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.77 + 2.77i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.86iT - 19T^{2} \) |
| 23 | \( 1 + (-3.75 + 3.75i)T - 23iT^{2} \) |
| 29 | \( 1 - 3.62T + 29T^{2} \) |
| 37 | \( 1 + (-1.42 + 1.42i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.12iT - 41T^{2} \) |
| 43 | \( 1 + (-6.90 - 6.90i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.33 - 5.33i)T + 47iT^{2} \) |
| 53 | \( 1 + (-7.93 + 7.93i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.98T + 59T^{2} \) |
| 61 | \( 1 + 3.54T + 61T^{2} \) |
| 67 | \( 1 + (-3.66 + 3.66i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.57iT - 71T^{2} \) |
| 73 | \( 1 + (-0.0629 - 0.0629i)T + 73iT^{2} \) |
| 79 | \( 1 - 13.3iT - 79T^{2} \) |
| 83 | \( 1 + (4.31 - 4.31i)T - 83iT^{2} \) |
| 89 | \( 1 - 5.00T + 89T^{2} \) |
| 97 | \( 1 + (-7.70 + 7.70i)T - 97iT^{2} \) |
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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
−10.52768122285223250219141791542, −9.211341612386207885686434124905, −8.228812987838903878268341228717, −7.31892915212053731052560320392, −6.39535101592095776777117932902, −6.13904696466156332078410866517, −5.01225505466745637748758920234, −4.07763847594041890796073569573, −2.72682082929841428973143497452, −1.16625158220418819067293424885,
1.19005303346506620781210641321, 2.28969164499006413269023151615, 3.98530224266395506909024078356, 4.81715576532043727528495025326, 5.42448255478317070135368388495, 6.14719700364086145374299169362, 7.27951790209779082902684994513, 8.695903208211495733519875373707, 9.262248553470778129105271594202, 10.30912318084809344780364434361