L(s) = 1 | − 0.655i·2-s + i·3-s + 1.57·4-s + 3.40i·5-s + 0.655·6-s − 2.33i·8-s − 9-s + 2.22·10-s + (1.48 + 2.96i)11-s + 1.57i·12-s + 5.44·13-s − 3.40·15-s + 1.60·16-s − 1.70·17-s + 0.655i·18-s + 6.25·19-s + ⋯ |
L(s) = 1 | − 0.463i·2-s + 0.577i·3-s + 0.785·4-s + 1.52i·5-s + 0.267·6-s − 0.827i·8-s − 0.333·9-s + 0.704·10-s + (0.448 + 0.893i)11-s + 0.453i·12-s + 1.51·13-s − 0.878·15-s + 0.402·16-s − 0.412·17-s + 0.154i·18-s + 1.43·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.293390244\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.293390244\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-1.48 - 2.96i)T \) |
good | 2 | \( 1 + 0.655iT - 2T^{2} \) |
| 5 | \( 1 - 3.40iT - 5T^{2} \) |
| 13 | \( 1 - 5.44T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 - 6.25T + 19T^{2} \) |
| 23 | \( 1 + 6.18T + 23T^{2} \) |
| 29 | \( 1 + 3.61iT - 29T^{2} \) |
| 31 | \( 1 - 6.50iT - 31T^{2} \) |
| 37 | \( 1 - 0.165T + 37T^{2} \) |
| 41 | \( 1 + 1.35T + 41T^{2} \) |
| 43 | \( 1 + 3.12iT - 43T^{2} \) |
| 47 | \( 1 - 0.800iT - 47T^{2} \) |
| 53 | \( 1 - 6.40T + 53T^{2} \) |
| 59 | \( 1 - 1.78iT - 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 + 8.89T + 67T^{2} \) |
| 71 | \( 1 - 3.59T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 4.73iT - 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 3.82iT - 89T^{2} \) |
| 97 | \( 1 - 3.87iT - 97T^{2} \) |
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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
−9.968978404927553878613839452436, −8.948041896735703817305131155809, −7.78843050951098903397062153898, −7.03278254479175899784732535678, −6.40878497998994861070978210715, −5.66507272586722577811478141596, −4.12032461526506476777402286230, −3.47491960098324144074366756735, −2.67625672425589544162423155730, −1.58562073120137576182791198913,
0.945448158143654190459566598022, 1.74412239932683414530043736764, 3.20613443833151561854696038670, 4.26857809892196789547145632917, 5.67819297677159342065292577192, 5.77092374378559383686866224952, 6.79947674702716522855227315998, 7.76788142240930395957563344737, 8.421034119927587931463080098408, 8.847752325149991503261072828318