| L(s) = 1 | + 1.09·2-s + (−0.323 − 1.70i)3-s − 0.791·4-s + (−0.355 − 1.87i)6-s + (−2.44 + i)7-s − 3.06·8-s + (−2.79 + 1.09i)9-s + 3.06i·11-s + (0.255 + 1.34i)12-s − 2.44·13-s + (−2.69 + 1.09i)14-s − 1.79·16-s − 2.69i·17-s + (−3.06 + 1.20i)18-s − 4.38i·19-s + ⋯ |
| L(s) = 1 | + 0.777·2-s + (−0.186 − 0.982i)3-s − 0.395·4-s + (−0.144 − 0.763i)6-s + (−0.925 + 0.377i)7-s − 1.08·8-s + (−0.930 + 0.366i)9-s + 0.925i·11-s + (0.0737 + 0.388i)12-s − 0.679·13-s + (−0.719 + 0.293i)14-s − 0.447·16-s − 0.653i·17-s + (−0.723 + 0.284i)18-s − 1.00i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0180371 + 0.0662091i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0180371 + 0.0662091i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.323 + 1.70i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.44 - i)T \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 11 | \( 1 - 3.06iT - 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + 2.69iT - 17T^{2} \) |
| 19 | \( 1 + 4.38iT - 19T^{2} \) |
| 23 | \( 1 + 5.26T + 23T^{2} \) |
| 29 | \( 1 - 5.26iT - 29T^{2} \) |
| 31 | \( 1 - 6.83iT - 31T^{2} \) |
| 37 | \( 1 + 8.58iT - 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 6.58iT - 43T^{2} \) |
| 47 | \( 1 + 2.69iT - 47T^{2} \) |
| 53 | \( 1 - 3.93T + 53T^{2} \) |
| 59 | \( 1 + 7.51T + 59T^{2} \) |
| 61 | \( 1 + 6.83iT - 61T^{2} \) |
| 67 | \( 1 + 4.16iT - 67T^{2} \) |
| 71 | \( 1 - 3.06iT - 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 0.582T + 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 - 7.51T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−10.33453891595398111384194995624, −9.340405394698076157677748159026, −8.641983777689555822925809734191, −7.26090087274404165551298059975, −6.68248416044971411516304060387, −5.56036708933646679483470320672, −4.82746614581677351517055474911, −3.36563784045836363327497205265, −2.28860016229234199131964939212, −0.03024089265334751831248067701,
2.99571335218248148935653505666, 3.81543942948081449809640609797, 4.57431006565948782083249876293, 5.88523823817509375097548069313, 6.17530349597120966085981964015, 7.960417865995671910241637759163, 8.857581125248805253635399086767, 9.931201958324314021385035679193, 10.15917372751401308919328654395, 11.56115269786078976502310047768
