| L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s − 4.60i·7-s − i·8-s + 9-s − 10-s + 12-s + 3.60·13-s + 4.60·14-s − i·15-s + 16-s + 4.60·17-s + i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 1.74i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s + 1.00·13-s + 1.23·14-s − 0.258i·15-s + 0.250·16-s + 1.11·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08592\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 - 3.60T \) |
| good | 7 | \( 1 + 4.60iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 + 4.60iT - 19T^{2} \) |
| 23 | \( 1 + 1.39T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 9.21iT - 37T^{2} \) |
| 41 | \( 1 - 3.21iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 9.21iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 9.21iT - 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 3.21iT - 67T^{2} \) |
| 71 | \( 1 - 9.21iT - 71T^{2} \) |
| 73 | \( 1 + 1.39iT - 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 2.78iT - 83T^{2} \) |
| 89 | \( 1 - 15.2iT - 89T^{2} \) |
| 97 | \( 1 - 1.39iT - 97T^{2} \) |
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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
−10.98590480735476491680464623781, −10.55545631564459147692087046345, −9.609856461462203484343119786176, −8.292324832698794270196394052586, −7.28408370850552558868117719050, −6.75684921745809903690553621907, −5.67092512818433186732684878306, −4.44528630915969865537365404958, −3.49828169818915795469979154788, −0.913082115461151326886198426371,
1.47051304109435739982787953074, 2.95706065298897152912152530358, 4.36571903843325057029973807942, 5.66855613071576167916315318375, 6.00583179444314314035526684452, 7.911395839490672226612383495233, 8.701653561132389674737517724002, 9.571349330763240661057203973055, 10.46115682256250730965007588703, 11.58732145091931953414428592509
