| L(s) = 1 | − 1.13·5-s + 35.8·7-s − 117.·11-s + 240. i·13-s − 147. i·17-s + 517. i·19-s − 184. i·23-s − 623.·25-s + 1.00e3·29-s − 458.·31-s − 40.6·35-s − 754. i·37-s − 940. i·41-s − 1.94e3i·43-s + 953. i·47-s + ⋯ |
| L(s) = 1 | − 0.0453·5-s + 0.732·7-s − 0.972·11-s + 1.42i·13-s − 0.511i·17-s + 1.43i·19-s − 0.348i·23-s − 0.997·25-s + 1.19·29-s − 0.477·31-s − 0.0332·35-s − 0.550i·37-s − 0.559i·41-s − 1.05i·43-s + 0.431i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2417272836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2417272836\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.13T + 625T^{2} \) |
| 7 | \( 1 - 35.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 117.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 240. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 147. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 517. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 184. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.00e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 458.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 754. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 940. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.94e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 953. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.97e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.52e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 418. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 426. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 6.39e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.99e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.62e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 5.13e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 7.57e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.04e3T + 8.85e7T^{2} \) |
| 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
−8.846642456463640609069778484851, −8.053777653380421240276962340248, −7.39860777961326544835529699414, −6.39396272548118162743501770177, −5.44256505923825875833132494455, −4.60340587386929512408495629216, −3.72166159927613150248632830162, −2.37556651083863864217127711630, −1.55327193278891857443903494496, −0.05038537803344893750973276053,
1.10993813597464692158107566783, 2.41849037736698785731420472173, 3.26900110580946197015622131508, 4.60225170720085738966096473211, 5.23741533522111014541683907397, 6.12577380407671743144976190471, 7.26785975187030901622154749880, 8.043315868132074733715679848683, 8.485462626639059349857025841785, 9.710580219746207128395803406098
