| L(s) = 1 | + (0.976 − 0.214i)2-s + (0.308 + 1.70i)3-s + (0.907 − 0.419i)4-s + (−0.897 − 0.249i)5-s + (0.668 + 1.59i)6-s + (3.72 − 3.53i)7-s + (0.796 − 0.605i)8-s + (−2.80 + 1.05i)9-s + (−0.929 − 0.0504i)10-s + (3.55 + 4.18i)11-s + (0.995 + 1.41i)12-s + (−0.168 − 0.499i)13-s + (2.88 − 4.25i)14-s + (0.147 − 1.60i)15-s + (0.647 − 0.762i)16-s + (−1.92 + 2.02i)17-s + ⋯ |
| L(s) = 1 | + (0.690 − 0.152i)2-s + (0.178 + 0.983i)3-s + (0.453 − 0.209i)4-s + (−0.401 − 0.111i)5-s + (0.272 + 0.652i)6-s + (1.40 − 1.33i)7-s + (0.281 − 0.213i)8-s + (−0.936 + 0.350i)9-s + (−0.294 − 0.0159i)10-s + (1.07 + 1.26i)11-s + (0.287 + 0.409i)12-s + (−0.0467 − 0.138i)13-s + (0.770 − 1.13i)14-s + (0.0380 − 0.414i)15-s + (0.161 − 0.190i)16-s + (−0.465 + 0.491i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.16401 + 0.383060i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16401 + 0.383060i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.976 + 0.214i)T \) |
| 3 | \( 1 + (-0.308 - 1.70i)T \) |
| 59 | \( 1 + (7.03 + 3.08i)T \) |
| good | 5 | \( 1 + (0.897 + 0.249i)T + (4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-3.72 + 3.53i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-3.55 - 4.18i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.168 + 0.499i)T + (-10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (1.92 - 2.02i)T + (-0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.0813 + 0.204i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (5.48 - 0.596i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (0.478 - 2.17i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (-7.77 - 3.09i)T + (22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (3.57 - 4.69i)T + (-9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (-1.17 + 10.7i)T + (-40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (6.17 + 5.24i)T + (6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (1.50 + 5.42i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (-5.07 + 0.275i)T + (52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (-1.90 - 8.65i)T + (-55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (2.77 + 3.65i)T + (-17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (15.5 - 4.30i)T + (60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (-0.536 - 0.363i)T + (27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (-0.328 - 2.00i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (-4.90 - 9.24i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (-6.74 - 1.48i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (5.54 - 3.76i)T + (35.9 - 90.1i)T^{2} \) |
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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
−11.68020352862123046695098036493, −10.47720345734400254708401903463, −10.14521282795046521094866197326, −8.683789868699964716785076242211, −7.76060852594037969111266389397, −6.72918750977809339293692465846, −5.14259828447726665211527532793, −4.26005614759094306919851091836, −3.89503811158888593465910149285, −1.83556960219543012481550111534,
1.71841790268522708604543313469, 2.95870986086463732409287181186, 4.44997465817363088531342644126, 5.78613406693107997957310220266, 6.34981062245423609994934818606, 7.78398340375064101272008021865, 8.304342157325655447776466983081, 9.200609035067507154838060442758, 11.19484654057249266041503828978, 11.74497722124522677494195367251
