L(s) = 1 | − 1.78·3-s + 1.23i·5-s + 2.64i·7-s + 0.171·9-s − 7.17i·13-s − 2.19i·15-s + 6.81·19-s − 4.71i·21-s + 7.48i·23-s + 3.48·25-s + 5.03·27-s − 3.25·35-s + 12.7i·39-s + 0.211i·45-s − 7.00·49-s + ⋯ |
L(s) = 1 | − 1.02·3-s + 0.550i·5-s + 0.999i·7-s + 0.0571·9-s − 1.98i·13-s − 0.565i·15-s + 1.56·19-s − 1.02i·21-s + 1.56i·23-s + 0.697·25-s + 0.969·27-s − 0.550·35-s + 2.04i·39-s + 0.0314i·45-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9490664395\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9490664395\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 5 | \( 1 - 1.23iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 7.17iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 6.81T + 19T^{2} \) |
| 23 | \( 1 - 7.48iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 10.6iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 15.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 5.29iT - 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.639929482257576732549306223106, −8.674017934446428751314945437270, −7.79587274779194794794362161996, −7.08346884707546189977457226816, −5.91833835412784692596437399504, −5.60092999007225429947545851796, −4.97375389313595013406809781704, −3.31049570588118957232065995648, −2.79290465102136854393894052598, −1.07056485266897956125749857677,
0.50120665552685616144983625922, 1.61938729458132757213472895836, 3.21465250860703987468641837063, 4.57842736870972473178908238661, 4.69071485298232189177657382405, 5.92724723604883075683917271831, 6.68625182698004782117763232958, 7.25134633915673382900896884472, 8.348998029181921871249698876823, 9.171987067671335148507357854534