L(s) = 1 | + (7.28 − 7.28i)5-s + 29.9·7-s + (0.408 + 0.408i)11-s + (26.6 − 26.6i)13-s + 83.0i·17-s + (51.6 + 51.6i)19-s + 173. i·23-s + 18.7i·25-s + (167. + 167. i)29-s + 191. i·31-s + (218. − 218. i)35-s + (−185. − 185. i)37-s − 62.7·41-s + (193. − 193. i)43-s + 93.1·47-s + ⋯ |
L(s) = 1 | + (0.651 − 0.651i)5-s + 1.61·7-s + (0.0111 + 0.0111i)11-s + (0.568 − 0.568i)13-s + 1.18i·17-s + (0.623 + 0.623i)19-s + 1.57i·23-s + 0.150i·25-s + (1.07 + 1.07i)29-s + 1.10i·31-s + (1.05 − 1.05i)35-s + (−0.823 − 0.823i)37-s − 0.239·41-s + (0.685 − 0.685i)43-s + 0.288·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.208556139\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.208556139\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-7.28 + 7.28i)T - 125iT^{2} \) |
| 7 | \( 1 - 29.9T + 343T^{2} \) |
| 11 | \( 1 + (-0.408 - 0.408i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-26.6 + 26.6i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 - 83.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-51.6 - 51.6i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 - 173. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-167. - 167. i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 - 191. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (185. + 185. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 62.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-193. + 193. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 - 93.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + (249. - 249. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-24.6 - 24.6i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (451. - 451. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (453. + 453. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 348. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 923. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 989. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-325. + 325. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 997.T + 9.12e5T^{2} \) |
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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
−9.307581189760688695096396073542, −8.606760530708987525180723300345, −7.995680705589902740686698744384, −7.12835031004941750968978455361, −5.66001618335597605179632439006, −5.44102512404658656839628775436, −4.38795863914687241680103947584, −3.28041945878947543077638741716, −1.57074039110600760437881744467, −1.38133357034166059104729789853,
0.823603605868425626200877175255, 2.04909895197309012719853584867, 2.83230404240383250075884702720, 4.39832970700126478742504202608, 4.93797211086662404368485072703, 6.09394480228387975281463719053, 6.81254853737503883541463529181, 7.79128260750734235118748381444, 8.491708077389187726897642962844, 9.390696323068416169870251263272