L(s) = 1 | + (0.707 − 0.707i)3-s + (1.28 − 1.82i)5-s + (1.75 − 1.97i)7-s − 1.00i·9-s + 2.67·11-s + (−1.22 + 1.22i)13-s + (−0.379 − 2.20i)15-s + (4.74 + 4.74i)17-s + 6.01·19-s + (−0.152 − 2.64i)21-s + (0.175 + 0.175i)23-s + (−1.67 − 4.71i)25-s + (−0.707 − 0.707i)27-s + 0.304i·29-s + 7.25i·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.576 − 0.816i)5-s + (0.665 − 0.746i)7-s − 0.333i·9-s + 0.805·11-s + (−0.340 + 0.340i)13-s + (−0.0979 − 0.568i)15-s + (1.15 + 1.15i)17-s + 1.38·19-s + (−0.0332 − 0.576i)21-s + (0.0366 + 0.0366i)23-s + (−0.334 − 0.942i)25-s + (−0.136 − 0.136i)27-s + 0.0566i·29-s + 1.30i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.628468176\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.628468176\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.28 + 1.82i)T \) |
| 7 | \( 1 + (-1.75 + 1.97i)T \) |
good | 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 + (1.22 - 1.22i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.74 - 4.74i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.01T + 19T^{2} \) |
| 23 | \( 1 + (-0.175 - 0.175i)T + 23iT^{2} \) |
| 29 | \( 1 - 0.304iT - 29T^{2} \) |
| 31 | \( 1 - 7.25iT - 31T^{2} \) |
| 37 | \( 1 + (0.735 - 0.735i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.05iT - 41T^{2} \) |
| 43 | \( 1 + (0.304 + 0.304i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.556 - 0.556i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.99 + 4.99i)T + 53iT^{2} \) |
| 59 | \( 1 + 7.98T + 59T^{2} \) |
| 61 | \( 1 - 5.53iT - 61T^{2} \) |
| 67 | \( 1 + (-3.43 + 3.43i)T - 67iT^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + (10.0 - 10.0i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.2iT - 79T^{2} \) |
| 83 | \( 1 + (4.88 - 4.88i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.91T + 89T^{2} \) |
| 97 | \( 1 + (-8.84 - 8.84i)T + 97iT^{2} \) |
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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
−9.113652889425262566616338964686, −8.454437991625453790103455102425, −7.65921548989232211042052090270, −6.96968852762326821981473091779, −5.92814476267023460375895759934, −5.11942504661229295076668311226, −4.18154183618513578307252095891, −3.25302881929121552535557613418, −1.68426389345896051208933749877, −1.17258642491509972691804441691,
1.46351429508349018506931017905, 2.70734530473096481851232177469, 3.26633614749008050068935470052, 4.59281120671920182281720451328, 5.46312973656750310720058812404, 6.10381676034149076382538479630, 7.34900779560541395836954138467, 7.75580883748965618998945175195, 8.891515113565930553847051921680, 9.649664456675196608220089494129