| L(s) = 1 | + (−3.30 + 5.73i)5-s + (−4.31 − 18.0i)7-s + (1.54 − 0.893i)11-s − 4.72i·13-s + (23.8 + 41.3i)17-s + (−40.1 − 23.1i)19-s + (30.2 + 17.4i)23-s + (40.5 + 70.3i)25-s + 48.3i·29-s + (−107. + 62.2i)31-s + (117. + 34.8i)35-s + (−137. + 238. i)37-s + 37.3·41-s − 215.·43-s + (−53.1 + 91.9i)47-s + ⋯ |
| L(s) = 1 | + (−0.295 + 0.512i)5-s + (−0.233 − 0.972i)7-s + (0.0424 − 0.0244i)11-s − 0.100i·13-s + (0.340 + 0.589i)17-s + (−0.484 − 0.279i)19-s + (0.273 + 0.158i)23-s + (0.324 + 0.562i)25-s + 0.309i·29-s + (−0.624 + 0.360i)31-s + (0.567 + 0.168i)35-s + (−0.612 + 1.06i)37-s + 0.142·41-s − 0.765·43-s + (−0.164 + 0.285i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8166017274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8166017274\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (4.31 + 18.0i)T \) |
| good | 5 | \( 1 + (3.30 - 5.73i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-1.54 + 0.893i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 4.72iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-23.8 - 41.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (40.1 + 23.1i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-30.2 - 17.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 48.3iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (107. - 62.2i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (137. - 238. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 37.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 215.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (53.1 - 91.9i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (233. - 134. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-149. - 259. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (292. + 168. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (188. + 326. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 816. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (596. - 344. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (307. - 531. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (480. - 832. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 449. iT - 9.12e5T^{2} \) |
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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
−10.73631804642907537856726442901, −10.13140441488139205716566627006, −9.042443986401982010177066980910, −7.989972279234140753371818486569, −7.13042123915680939771200753206, −6.41655591181320119705616263946, −5.09159653399775436887840363007, −3.91649883617194946407777508961, −3.05374150975776621314529498022, −1.34605133789120566019324550934,
0.25969470102305050995711045531, 1.95840890714136022190899746827, 3.23051317354448071760924234509, 4.52148008181840809325683322106, 5.47817999987433358985486477343, 6.43208218601876260890137597950, 7.57876166994684866462209506370, 8.566495823285626772467155789136, 9.173396529437101444865110880854, 10.12990354788855438944706548866
