L(s) = 1 | + 5.61i·3-s − 2.98i·5-s − 6.83i·7-s − 22.5·9-s − 6.88·11-s + 12.4·13-s + 16.7·15-s − 15.8·17-s − 19.2i·19-s + 38.3·21-s + 33.2·23-s + 16.0·25-s − 76.1i·27-s − 45.8i·29-s − 14.7·31-s + ⋯ |
L(s) = 1 | + 1.87i·3-s − 0.597i·5-s − 0.975i·7-s − 2.50·9-s − 0.626·11-s + 0.953·13-s + 1.11·15-s − 0.930·17-s − 1.01i·19-s + 1.82·21-s + 1.44·23-s + 0.642·25-s − 2.82i·27-s − 1.58i·29-s − 0.475·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.329556858\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.329556858\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 43 | \( 1 + (-40.8 - 13.3i)T \) |
good | 3 | \( 1 - 5.61iT - 9T^{2} \) |
| 5 | \( 1 + 2.98iT - 25T^{2} \) |
| 7 | \( 1 + 6.83iT - 49T^{2} \) |
| 11 | \( 1 + 6.88T + 121T^{2} \) |
| 13 | \( 1 - 12.4T + 169T^{2} \) |
| 17 | \( 1 + 15.8T + 289T^{2} \) |
| 19 | \( 1 + 19.2iT - 361T^{2} \) |
| 23 | \( 1 - 33.2T + 529T^{2} \) |
| 29 | \( 1 + 45.8iT - 841T^{2} \) |
| 31 | \( 1 + 14.7T + 961T^{2} \) |
| 37 | \( 1 + 13.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 2.49T + 1.68e3T^{2} \) |
| 47 | \( 1 - 10.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 31.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 64.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 78.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 89.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 35.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 50.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 10.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 13.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 66.6T + 9.40e3T^{2} \) |
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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
−10.32090820696979235142580552288, −9.266485595984761778635236092674, −8.913441070511483958739195516218, −7.87030289996590869151583384534, −6.51758090578239721745847638525, −5.31178635634272454785909645984, −4.59931543981958564065294772380, −3.93298601390195353200186982962, −2.81158146645241148601081952703, −0.51218581229114630577449401835,
1.27140451336358858423526151304, 2.39298100985171063675837955970, 3.20913642521113627050790991176, 5.26636239408128263515256631613, 6.08330383427982316352246653651, 6.81882761737854906327382211924, 7.53444668517534458856338928020, 8.570574044189909799332650334224, 8.963035020878450627930505372526, 10.73393736472804450694621443480