| L(s) = 1 | + (−1.41 + 0.0466i)2-s + (−1.17 − 1.39i)3-s + (1.99 − 0.131i)4-s + (−1.08 + 1.95i)5-s + (1.72 + 1.92i)6-s + (1.36 − 2.36i)7-s + (−2.81 + 0.279i)8-s + (−0.0573 + 0.325i)9-s + (1.44 − 2.81i)10-s + (−2.24 + 1.29i)11-s + (−2.52 − 2.63i)12-s + (−0.527 − 0.443i)13-s + (−1.82 + 3.41i)14-s + (4.00 − 0.777i)15-s + (3.96 − 0.526i)16-s + (0.544 − 0.0960i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0329i)2-s + (−0.677 − 0.807i)3-s + (0.997 − 0.0659i)4-s + (−0.484 + 0.874i)5-s + (0.703 + 0.784i)6-s + (0.517 − 0.895i)7-s + (−0.995 + 0.0987i)8-s + (−0.0191 + 0.108i)9-s + (0.455 − 0.890i)10-s + (−0.675 + 0.389i)11-s + (−0.728 − 0.760i)12-s + (−0.146 − 0.122i)13-s + (−0.487 + 0.912i)14-s + (1.03 − 0.200i)15-s + (0.991 − 0.131i)16-s + (0.132 − 0.0232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00530705 - 0.200581i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00530705 - 0.200581i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0466i)T \) |
| 5 | \( 1 + (1.08 - 1.95i)T \) |
| 19 | \( 1 + (1.26 + 4.16i)T \) |
| good | 3 | \( 1 + (1.17 + 1.39i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (-1.36 + 2.36i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.24 - 1.29i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.527 + 0.443i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.544 + 0.0960i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.366 + 0.133i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (8.63 + 1.52i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (4.91 - 8.51i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.48T + 37T^{2} \) |
| 41 | \( 1 + (3.19 + 3.81i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.46 + 0.898i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.347 - 1.97i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-9.32 + 3.39i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.22 + 6.94i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.23 - 0.449i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-3.08 - 0.543i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (3.62 + 1.31i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (5.31 + 6.33i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-9.44 + 7.92i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.643 - 1.11i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.29 + 8.69i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.36 - 7.75i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.79578013904853396496273253031, −10.35123119022200955567263279615, −9.004322755830811342258070255386, −7.73655831455356387838899043036, −7.21809169915301678666672594682, −6.65324361651699261859821238819, −5.30340484586303639515081802942, −3.49329734927512076826447711636, −1.88756118670957902704294614280, −0.19018798271175371194260117962,
1.95829574573404776883481915982, 3.81973380420924143691226602681, 5.29040787537540002755395336823, 5.73625733947728850719018309440, 7.48685577791518097883905164928, 8.277604134590174648445497772850, 9.054959288759025160110999977835, 9.965320872568169162650912669671, 10.87985736144813931560579835420, 11.56611159683383098688677524550
