| L(s) = 1 | − 2.44i·3-s − 1.66i·5-s + 3.65i·7-s − 2.98·9-s + 1.95i·11-s − 1.88·13-s − 4.06·15-s + (−3.75 − 1.70i)17-s + 4.87·19-s + 8.95·21-s − 9.29i·23-s + 2.24·25-s − 0.0283i·27-s + 3.33i·29-s − 4.53i·31-s + ⋯ |
| L(s) = 1 | − 1.41i·3-s − 0.742i·5-s + 1.38i·7-s − 0.996·9-s + 0.588i·11-s − 0.522·13-s − 1.04·15-s + (−0.910 − 0.414i)17-s + 1.11·19-s + 1.95·21-s − 1.93i·23-s + 0.448·25-s − 0.00545i·27-s + 0.619i·29-s − 0.814i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7562250831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7562250831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.75 + 1.70i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 5 | \( 1 + 1.66iT - 5T^{2} \) |
| 7 | \( 1 - 3.65iT - 7T^{2} \) |
| 11 | \( 1 - 1.95iT - 11T^{2} \) |
| 13 | \( 1 + 1.88T + 13T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 23 | \( 1 + 9.29iT - 23T^{2} \) |
| 29 | \( 1 - 3.33iT - 29T^{2} \) |
| 31 | \( 1 + 4.53iT - 31T^{2} \) |
| 37 | \( 1 + 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 0.268iT - 41T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 - 2.67T + 53T^{2} \) |
| 61 | \( 1 + 2.27iT - 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 6.53iT - 71T^{2} \) |
| 73 | \( 1 + 6.29iT - 73T^{2} \) |
| 79 | \( 1 - 2.53iT - 79T^{2} \) |
| 83 | \( 1 + 2.22T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−8.135335612025995629660813270377, −7.18071467930758307139790673531, −6.70645181772435411528392353119, −5.89527099644501096167122027225, −5.08927912846338022028629774713, −4.50435966376148533729043990440, −2.87702610570439635435561544214, −2.30109783221173992880633128233, −1.45262428707370057215225509475, −0.21380500750965348412775535946,
1.43025359673623382901716705129, 3.09052667312591000501715359166, 3.43269371496130352530770128099, 4.27438273032090991204537979486, 4.94388387155546336636689979143, 5.75760675732091237355857216126, 6.82206440896657912681570972513, 7.28798830851253065356912933361, 8.147778584532507818679740824708, 9.059227544843703990384131419732
