| L(s) = 1 | + (0.766 + 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (0.0262 − 0.148i)5-s + (0.173 + 0.984i)6-s + (2.53 + 0.769i)7-s + (−0.500 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.115 − 0.0970i)10-s + (−0.166 − 0.288i)11-s + (−0.500 + 0.866i)12-s + (−0.392 − 2.22i)13-s + (1.44 + 2.21i)14-s + (0.115 − 0.0970i)15-s + (−0.939 + 0.342i)16-s + (−0.929 + 5.27i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.442 + 0.371i)3-s + (0.0868 + 0.492i)4-s + (0.0117 − 0.0664i)5-s + (0.0708 + 0.402i)6-s + (0.956 + 0.290i)7-s + (−0.176 + 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.0365 − 0.0306i)10-s + (−0.0502 − 0.0870i)11-s + (−0.144 + 0.249i)12-s + (−0.108 − 0.617i)13-s + (0.386 + 0.592i)14-s + (0.0298 − 0.0250i)15-s + (−0.234 + 0.0855i)16-s + (−0.225 + 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98264 + 1.69556i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98264 + 1.69556i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (-2.53 - 0.769i)T \) |
| 19 | \( 1 + (-4.07 - 1.53i)T \) |
| good | 5 | \( 1 + (-0.0262 + 0.148i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.166 + 0.288i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.392 + 2.22i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.929 - 5.27i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-3.88 - 1.41i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (5.92 + 2.15i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + 4.22T + 31T^{2} \) |
| 37 | \( 1 + (5.97 + 10.3i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.300 + 1.70i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (2.15 + 1.80i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.10 - 6.25i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.09 - 6.22i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.19 + 6.79i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.65 - 0.601i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.88 + 2.42i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (7.21 + 6.05i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.09 - 1.75i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-9.12 + 3.32i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.97 + 6.89i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.32 - 2.78i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (3.87 - 1.41i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.69506241137462240772053456528, −9.376096771738811344022730802915, −8.677470604200499306155214953610, −7.84224599805704919769927836157, −7.18454838528042269790432595037, −5.72744577891120257968761853767, −5.23670566367972539887635795876, −4.11726009439451144772724029723, −3.19140522383497542485493161213, −1.80997649033581619493675304317,
1.20143071154571908480514551482, 2.40919698070998953463429355538, 3.46127091064745232184948854408, 4.71918142217065207404615592836, 5.28569323641639113855042966181, 6.86637456112754000199776597541, 7.24449152847039370959712113543, 8.475808134277701850810107560577, 9.229884712048263752819000892688, 10.14774557471258113323159643167
