| L(s) = 1 | + 2.82·2-s + 8.00·4-s + 23.9i·5-s + 22.6·8-s + 67.7i·10-s − 97.9·11-s − 104. i·13-s + 64.0·16-s + 107. i·17-s + 38.3i·19-s + 191. i·20-s − 277.·22-s + 1.02e3·23-s + 51.3·25-s − 294. i·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.958i·5-s + 0.353·8-s + 0.677i·10-s − 0.809·11-s − 0.616i·13-s + 0.250·16-s + 0.372i·17-s + 0.106i·19-s + 0.479i·20-s − 0.572·22-s + 1.93·23-s + 0.0821·25-s − 0.436i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.027811788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.027811788\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 23.9iT - 625T^{2} \) |
| 11 | \( 1 + 97.9T + 1.46e4T^{2} \) |
| 13 | \( 1 + 104. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 107. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 38.3iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 1.02e3T + 2.79e5T^{2} \) |
| 29 | \( 1 + 621.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.51e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 562.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.02e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.38e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.94e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.19e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.93e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 665. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 5.92e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.49e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 8.96e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 1.04e4T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.26e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 3.60e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.95e3iT - 8.85e7T^{2} \) |
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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
−10.27496441621019789031394723858, −9.047141784973236344842802315857, −7.986495534047138720151668167850, −7.16545765837498663311843486048, −6.47568303529374184746926331521, −5.43990931699374404851198113059, −4.69266564214631298985402755086, −3.21020562357548848973775870257, −2.92128188646139892775562600578, −1.43599821836200695973862841225,
0.32472216775137108153385741475, 1.62008134584695543201234750783, 2.78272318655122574892598861457, 3.92978791071410520279288609486, 4.99528498466812057821760324589, 5.33942902716893952291199111538, 6.63127122517899256033757818895, 7.41422939649622049881588460592, 8.440917306978671868062379968220, 9.175927716402046312727008851724
