| L(s) = 1 | − 5.58i·2-s − 15.1·4-s + 29.7·5-s − 12.9i·7-s − 4.47i·8-s − 166. i·10-s + (100. − 67.4i)11-s − 36.6i·13-s − 72.2·14-s − 268.·16-s − 464. i·17-s + 327. i·19-s − 452.·20-s + (−376. − 560. i)22-s − 396.·23-s + ⋯ |
| L(s) = 1 | − 1.39i·2-s − 0.949·4-s + 1.18·5-s − 0.264i·7-s − 0.0698i·8-s − 1.66i·10-s + (0.830 − 0.557i)11-s − 0.216i·13-s − 0.368·14-s − 1.04·16-s − 1.60i·17-s + 0.907i·19-s − 1.13·20-s + (−0.778 − 1.15i)22-s − 0.750·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.594768 - 1.95163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.594768 - 1.95163i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-100. + 67.4i)T \) |
| good | 2 | \( 1 + 5.58iT - 16T^{2} \) |
| 5 | \( 1 - 29.7T + 625T^{2} \) |
| 7 | \( 1 + 12.9iT - 2.40e3T^{2} \) |
| 13 | \( 1 + 36.6iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 464. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 327. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 396.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.15e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 437.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 276.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.78e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.66e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.85e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 3.74e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 6.57e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 5.14e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.46e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 6.03e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 3.58e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 9.71e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.18e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.58e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.55e3T + 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.53657476877603933957835118693, −11.67810841606512844218031042094, −10.56406798994037047489705203928, −9.778439598784604789134142662209, −8.896324275588366881534936985945, −6.93838429013724390265067031805, −5.52399967961049037839264981482, −3.79201972378838421827052183321, −2.37040553992349012104497503666, −1.01253933285817067290898274475,
2.01516208387628016414794943823, 4.51551121523287369639729200704, 5.98986938371242841892265630114, 6.45828847269572532242287242660, 7.919050194894713336061765569571, 9.059717081285074237125296068076, 9.989414683091584891619606121879, 11.52818620182638639626794539265, 12.96007446759613489049607780073, 13.89779257288496808906339244799
