| L(s) = 1 | − 46.7i·3-s + 806. i·5-s + (−973. + 2.19e3i)7-s − 2.18e3·9-s − 2.99e3·11-s − 2.14e4i·13-s + 3.77e4·15-s + 8.15e4i·17-s − 1.82e5i·19-s + (1.02e5 + 4.55e4i)21-s − 4.49e5·23-s − 2.60e5·25-s + 1.02e5i·27-s − 4.74e5·29-s − 1.72e6i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.29i·5-s + (−0.405 + 0.914i)7-s − 0.333·9-s − 0.204·11-s − 0.752i·13-s + 0.745·15-s + 0.976i·17-s − 1.40i·19-s + (0.527 + 0.234i)21-s − 1.60·23-s − 0.667·25-s + 0.192i·27-s − 0.670·29-s − 1.86i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.381447121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381447121\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| 7 | \( 1 + (973. - 2.19e3i)T \) |
| good | 5 | \( 1 - 806. iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 2.99e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + 2.14e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 8.15e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.82e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 4.49e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 4.74e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.72e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 4.37e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.12e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 4.67e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 5.46e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 8.28e5T + 6.22e13T^{2} \) |
| 59 | \( 1 - 9.00e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.38e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 1.60e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.27e6T + 6.45e14T^{2} \) |
| 73 | \( 1 - 7.58e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.67e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 2.05e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.22e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.23e7iT - 7.83e15T^{2} \) |
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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
−10.22355870908214508425068176167, −9.186189917498885920005732343018, −8.066879138884001615155019648106, −7.25696001553662421452799979213, −6.20146241383646734332579404111, −5.68370596096134579236965701245, −3.91099983971094288993575893626, −2.71299811042533714025853365879, −2.18335554535894596007604211434, −0.40257859298092116773991510862,
0.66668141366314367664477493746, 1.79632017639252140877840703459, 3.48567109667295450351467758149, 4.32776403103534957907734450983, 5.15560557532312770815166908697, 6.28588292122352906533549943288, 7.56109919795689525044045318652, 8.460518102380047621328265617797, 9.459596335762321643294690472281, 10.00780416631086856229693648033
