| L(s) = 1 | + (1.72 − 3.37i)2-s + (−2.24 + 0.356i)3-s + (−6.10 − 8.39i)4-s + (−2.66 + 8.21i)6-s + (−0.264 − 0.0418i)7-s + (−23.9 + 3.78i)8-s + (−3.62 + 1.17i)9-s + (−8.77 − 6.62i)11-s + (16.7 + 16.7i)12-s + (16.3 + 8.31i)13-s + (−0.596 + 0.821i)14-s + (−15.5 + 47.8i)16-s + (−25.9 + 13.2i)17-s + (−2.26 + 14.2i)18-s + (12.9 − 17.8i)19-s + ⋯ |
| L(s) = 1 | + (0.860 − 1.68i)2-s + (−0.749 + 0.118i)3-s + (−1.52 − 2.09i)4-s + (−0.444 + 1.36i)6-s + (−0.0377 − 0.00598i)7-s + (−2.98 + 0.473i)8-s + (−0.403 + 0.131i)9-s + (−0.797 − 0.602i)11-s + (1.39 + 1.39i)12-s + (1.25 + 0.639i)13-s + (−0.0426 + 0.0586i)14-s + (−0.970 + 2.98i)16-s + (−1.52 + 0.778i)17-s + (−0.125 + 0.794i)18-s + (0.682 − 0.939i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0383 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0383 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.412642 + 0.397098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.412642 + 0.397098i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (8.77 + 6.62i)T \) |
| good | 2 | \( 1 + (-1.72 + 3.37i)T + (-2.35 - 3.23i)T^{2} \) |
| 3 | \( 1 + (2.24 - 0.356i)T + (8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (0.264 + 0.0418i)T + (46.6 + 15.1i)T^{2} \) |
| 13 | \( 1 + (-16.3 - 8.31i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (25.9 - 13.2i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-12.9 + 17.8i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (11.3 - 11.3i)T - 529iT^{2} \) |
| 29 | \( 1 + (20.5 + 28.3i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (1.29 + 3.99i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (35.4 + 5.61i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (18.8 + 13.7i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-28.8 + 28.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-6.00 - 37.9i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (40.7 + 20.7i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (21.5 + 29.6i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-12.2 + 37.7i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (51.4 + 51.4i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + (21.8 - 67.1i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-9.93 + 62.7i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-26.1 + 8.50i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-47.2 - 92.7i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + 6.15iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (0.706 - 1.38i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.11384667214770918150553722894, −10.67260145176722567146496348792, −9.408024781419694868216451312897, −8.484999499112560819902966969545, −6.31848979904331135860817340306, −5.52867294776856630364437687895, −4.49444230260044933911824531159, −3.37371258931947155282603855226, −1.99998944435132484339475256469, −0.23025391848922573148172853089,
3.27932349882507969182239567315, 4.66631637710579887321708244259, 5.53263503487401573192614020846, 6.27860463720495862587582400448, 7.19400023919173621008236816751, 8.201692257163450752426100592789, 9.057635161651607229180564478960, 10.63794923367490822695535687333, 11.77499157722745030525213713230, 12.70377566274386720356513211392
