| L(s) = 1 | − i·2-s − 3-s − 4-s + 1.73i·5-s + i·6-s − 1.26i·7-s + i·8-s + 9-s + 1.73·10-s − 1.26i·11-s + 12-s − 1.26·14-s − 1.73i·15-s + 16-s − 5.19·17-s − i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.774i·5-s + 0.408i·6-s − 0.479i·7-s + 0.353i·8-s + 0.333·9-s + 0.547·10-s − 0.382i·11-s + 0.288·12-s − 0.338·14-s − 0.447i·15-s + 0.250·16-s − 1.26·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.061344076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061344076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 + 1.26iT - 7T^{2} \) |
| 11 | \( 1 + 1.26iT - 11T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 4.73iT - 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 9.46iT - 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 6.46iT - 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 - 4.73iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 13.8iT - 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 7.26iT - 67T^{2} \) |
| 71 | \( 1 + 2.19iT - 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 - 5.66iT - 83T^{2} \) |
| 89 | \( 1 + 9.46iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−10.39929971212634421016208438997, −9.304277378374203918096418043511, −8.523036629893775872043002053990, −7.28859929299126824852460394202, −6.69010141680569072618755070945, −5.62935354024787477675599788695, −4.61998220779201820251734985220, −3.63435942592410133375442644641, −2.63377374998890150958925067877, −1.15608992486730674946899329763,
0.62353040943191588381020854108, 2.35480109575313929094526462624, 4.06580538218101753146201116042, 4.94674985340704599099193371863, 5.46976260484627359185834123977, 6.63781016951225988056677613452, 7.16120071605056755413715380335, 8.356451638229374534841972439635, 9.058367178104676738618115043563, 9.578592690548032495096346078703
