| L(s) = 1 | + 2.23i·2-s − 3.00·4-s + i·7-s − 2.23i·8-s − 6.47·11-s − 4.47i·13-s − 2.23·14-s − 0.999·16-s + 2i·17-s + 2.47·19-s − 14.4i·22-s + 4i·23-s + 10.0·26-s − 3.00i·28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 1.58i·2-s − 1.50·4-s + 0.377i·7-s − 0.790i·8-s − 1.95·11-s − 1.24i·13-s − 0.597·14-s − 0.249·16-s + 0.485i·17-s + 0.567·19-s − 3.08i·22-s + 0.834i·23-s + 1.96·26-s − 0.566i·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3311752022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3311752022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 - 2.23iT - 2T^{2} \) |
| 11 | \( 1 + 6.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 + 6.94iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94iT - 43T^{2} \) |
| 47 | \( 1 + 12.9iT - 47T^{2} \) |
| 53 | \( 1 + 3.52iT - 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 - 12.4iT - 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 16.9iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 8.47iT - 97T^{2} \) |
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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
−8.995463345681164203966507965678, −8.283319656894834948865818255447, −7.65568806226039765124669836481, −7.20138232855918643862526336413, −5.87063825595099053340774904441, −5.53079713591420088921059849880, −4.89732556690609095038396281743, −3.49030236416977741467500677624, −2.34102060443871521469960945171, −0.12944883863079460234655042444,
1.33320315469056512678066031339, 2.54547368211328641730225610703, 3.09391111446318855459571571870, 4.42666281802822174247804346152, 4.83226894706813712562199591532, 6.13826055286665482763449939279, 7.26006545338955513933159122230, 8.032974947596132428037322360716, 9.036975233631919605288719504541, 9.755900406247042820269731580961
