| L(s) = 1 | + (0.948 − 1.04i)2-s + (0.578 − 0.825i)3-s + (−0.200 − 1.98i)4-s + (0.127 − 1.45i)5-s + (−0.317 − 1.38i)6-s + (0.743 − 1.28i)7-s + (−2.27 − 1.67i)8-s + (0.678 + 1.86i)9-s + (−1.40 − 1.51i)10-s + (−0.782 + 2.91i)11-s + (−1.75 − 0.984i)12-s + (0.820 + 1.17i)13-s + (−0.646 − 2.00i)14-s + (−1.12 − 0.946i)15-s + (−3.91 + 0.799i)16-s + (−0.707 − 0.257i)17-s + ⋯ |
| L(s) = 1 | + (0.670 − 0.741i)2-s + (0.333 − 0.476i)3-s + (−0.100 − 0.994i)4-s + (0.0569 − 0.650i)5-s + (−0.129 − 0.567i)6-s + (0.281 − 0.487i)7-s + (−0.805 − 0.592i)8-s + (0.226 + 0.621i)9-s + (−0.444 − 0.478i)10-s + (−0.235 + 0.880i)11-s + (−0.507 − 0.284i)12-s + (0.227 + 0.325i)13-s + (−0.172 − 0.535i)14-s + (−0.291 − 0.244i)15-s + (−0.979 + 0.199i)16-s + (−0.171 − 0.0624i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08211 - 1.67117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08211 - 1.67117i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.948 + 1.04i)T \) |
| 19 | \( 1 + (3.27 + 2.87i)T \) |
| good | 3 | \( 1 + (-0.578 + 0.825i)T + (-1.02 - 2.81i)T^{2} \) |
| 5 | \( 1 + (-0.127 + 1.45i)T + (-4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-0.743 + 1.28i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.782 - 2.91i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.820 - 1.17i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.707 + 0.257i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.43 - 2.04i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.970 + 2.08i)T + (-18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (-3.54 + 6.13i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.41 - 4.41i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.593 - 3.36i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (11.2 + 0.988i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-0.0302 - 0.0829i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.837 - 9.57i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (0.747 - 0.348i)T + (37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (0.298 + 3.41i)T + (-60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (7.92 + 3.69i)T + (43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-5.87 - 7.00i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (2.48 - 0.437i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.43 + 8.12i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.56 + 9.58i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.413 - 2.34i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (3.31 - 9.10i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−11.52387802969408801277103250180, −10.64710124335118348219293489351, −9.718067062630214351644203657749, −8.677701145763998986072249008628, −7.52735780214041497816544470991, −6.44483709315306817223886255733, −4.91388636397324956779190897204, −4.37987815613198318725094279127, −2.60860085397421174161488706144, −1.39467955149879763805715441859,
2.82304805917463841787511124589, 3.72604304720833532152662352482, 5.01141316825776538308391160920, 6.15602170013345200976105868038, 6.92086338903539994586271443956, 8.353617445464717554113515189593, 8.782654802037947702597805013862, 10.19380631525602646177557606099, 11.13530936627932625813542543103, 12.19886784042433863118227791177
