| L(s) = 1 | − 2.82i·2-s − 8.00·4-s − 11.1i·5-s + 90.5·7-s + 22.6i·8-s − 31.6·10-s + 40.6i·11-s + 65.0·13-s − 256. i·14-s + 64.0·16-s − 555. i·17-s + 178.·19-s + 89.4i·20-s + 114.·22-s + 160. i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.447i·5-s + 1.84·7-s + 0.353i·8-s − 0.316·10-s + 0.335i·11-s + 0.384·13-s − 1.30i·14-s + 0.250·16-s − 1.92i·17-s + 0.495·19-s + 0.223i·20-s + 0.237·22-s + 0.302i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.783168705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.783168705\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 11.1iT \) |
| good | 7 | \( 1 - 90.5T + 2.40e3T^{2} \) |
| 11 | \( 1 - 40.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 65.0T + 2.85e4T^{2} \) |
| 17 | \( 1 + 555. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 178.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 160. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.15e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 419.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.11e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.18e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.66e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 979. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.07e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 4.60e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.61e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.13e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 8.24e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.50e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.01e4T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.65e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.25e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 8.34e3T + 8.85e7T^{2} \) |
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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
−9.325712794083312577601976704183, −8.827692318740353581111565532306, −7.82998791628023291215358778257, −7.17104957844516942642635159909, −5.42235650709826281179694304314, −4.97814123299649130070404570870, −4.08566220594932164955160703450, −2.70802568242155769898445825050, −1.60921944720194858383879911284, −0.78480046726764449212592029177,
1.04067624113918229168350888625, 2.14644189154138819586819795143, 3.76242079651472477579925384807, 4.57051605865620117029677851138, 5.61978584187224335919821700299, 6.31053494365841802251158598628, 7.49326240871990241908729741317, 8.178145008810835669944821487496, 8.606587257587477507641939239907, 9.891195786258436509139099616938
