| L(s) = 1 | + 2.99·3-s − 103.·5-s − 234.·9-s + 233.·11-s − 204.·13-s − 308.·15-s − 112.·17-s − 2.48e3·19-s − 3.34e3·23-s + 7.48e3·25-s − 1.42e3·27-s − 4.55e3·29-s − 8.97e3·31-s + 698.·33-s + 3.78e3·37-s − 610.·39-s + 3.65e3·41-s + 2.11e4·43-s + 2.41e4·45-s + 7.79e3·47-s − 335.·51-s − 9.87e3·53-s − 2.40e4·55-s − 7.44e3·57-s + 2.07e4·59-s + 4.44e4·61-s + 2.10e4·65-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 1.84·5-s − 0.963·9-s + 0.581·11-s − 0.334·13-s − 0.353·15-s − 0.0940·17-s − 1.58·19-s − 1.31·23-s + 2.39·25-s − 0.376·27-s − 1.00·29-s − 1.67·31-s + 0.111·33-s + 0.454·37-s − 0.0643·39-s + 0.339·41-s + 1.74·43-s + 1.77·45-s + 0.514·47-s − 0.0180·51-s − 0.482·53-s − 1.07·55-s − 0.303·57-s + 0.775·59-s + 1.52·61-s + 0.617·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6006939702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6006939702\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 2.99T + 243T^{2} \) |
| 5 | \( 1 + 103.T + 3.12e3T^{2} \) |
| 11 | \( 1 - 233.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 204.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 112.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.48e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.65e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.79e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.87e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.44e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.37e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.70e5T + 8.58e9T^{2} \) |
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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
−10.89510854046734961717584053963, −9.350959341407567517066036953459, −8.480884139852316502396920142080, −7.84391115469404763043039090846, −6.91336546993878244070952976493, −5.68286875816524345941118314163, −4.20836142591028421356321375023, −3.71266464456169246272345559098, −2.30785117780995352126246963504, −0.38612192193303480728592034268,
0.38612192193303480728592034268, 2.30785117780995352126246963504, 3.71266464456169246272345559098, 4.20836142591028421356321375023, 5.68286875816524345941118314163, 6.91336546993878244070952976493, 7.84391115469404763043039090846, 8.480884139852316502396920142080, 9.350959341407567517066036953459, 10.89510854046734961717584053963
