| L(s) = 1 | − 0.554i·3-s + 4.57·5-s + 8.69·9-s − 15.7i·11-s − 8.57·13-s − 2.53i·15-s − 28.3·17-s + 6.33i·19-s − 31.0i·23-s − 4.05·25-s − 9.81i·27-s − 0.846·29-s − 21.6i·31-s − 8.72·33-s − 33.6·37-s + ⋯ |
| L(s) = 1 | − 0.184i·3-s + 0.915·5-s + 0.965·9-s − 1.43i·11-s − 0.659·13-s − 0.169i·15-s − 1.66·17-s + 0.333i·19-s − 1.35i·23-s − 0.162·25-s − 0.363i·27-s − 0.0291·29-s − 0.697i·31-s − 0.264·33-s − 0.909·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.417976543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.417976543\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.554iT - 9T^{2} \) |
| 5 | \( 1 - 4.57T + 25T^{2} \) |
| 11 | \( 1 + 15.7iT - 121T^{2} \) |
| 13 | \( 1 + 8.57T + 169T^{2} \) |
| 17 | \( 1 + 28.3T + 289T^{2} \) |
| 19 | \( 1 - 6.33iT - 361T^{2} \) |
| 23 | \( 1 + 31.0iT - 529T^{2} \) |
| 29 | \( 1 + 0.846T + 841T^{2} \) |
| 31 | \( 1 + 21.6iT - 961T^{2} \) |
| 37 | \( 1 + 33.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 66.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 44.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 38.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 14.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 5.80iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 52.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 117. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 81.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 57.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 89.2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 3.44T + 9.40e3T^{2} \) |
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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
−8.861405436182769077716494932408, −8.348578312094751974678352105325, −7.15295541743384426413276870648, −6.52295340611120300421101377209, −5.78049541010834309579044001527, −4.80537757455152907045380065603, −3.87981008623521143075863905010, −2.58497371300746858109344940449, −1.75368601987525874108175042437, −0.34521240791121183847916361841,
1.66956178662246377904873872104, 2.20164602347276509339117555496, 3.67466902335910143856979817986, 4.76325335560455259807954518245, 5.15289435245436952649344662920, 6.57610889410904127921217144641, 6.94266315911979745257249798693, 7.84116921081069969425780529891, 9.069048893046388979785655514704, 9.601090603998181481157104011030
