L(s) = 1 | + (0.707 + 0.707i)2-s + (−1.70 − 0.292i)3-s + 1.00i·4-s + (−0.707 − 2.12i)5-s + (−0.999 − 1.41i)6-s + (−1 + i)7-s + (−0.707 + 0.707i)8-s + (2.82 + i)9-s + (0.999 − 2i)10-s + 1.41i·11-s + (0.292 − 1.70i)12-s − 1.41·14-s + (0.585 + 3.82i)15-s − 1.00·16-s + (1.41 + 1.41i)17-s + (1.29 + 2.70i)18-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.985 − 0.169i)3-s + 0.500i·4-s + (−0.316 − 0.948i)5-s + (−0.408 − 0.577i)6-s + (−0.377 + 0.377i)7-s + (−0.250 + 0.250i)8-s + (0.942 + 0.333i)9-s + (0.316 − 0.632i)10-s + 0.426i·11-s + (0.0845 − 0.492i)12-s − 0.377·14-s + (0.151 + 0.988i)15-s − 0.250·16-s + (0.342 + 0.342i)17-s + (0.304 + 0.638i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.656840 + 0.127568i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.656840 + 0.127568i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (1.70 + 0.292i)T \) |
| 5 | \( 1 + (0.707 + 2.12i)T \) |
good | 7 | \( 1 + (1 - i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + (-6 + 6i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-6 - 6i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (2.82 - 2.82i)T - 53iT^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (4 - 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (5 + 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 - 8.48i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
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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
−16.87828934394040068244633700629, −16.13159047983139062741175139356, −15.03063705647755594215940573671, −13.08387053589197631022679088028, −12.50343930246193362992094011565, −11.23367415944945866485573438577, −9.281829984004021987354363144779, −7.50373221530796780404781154817, −5.90669421672407264250503282656, −4.56412421266750816989800627129,
3.72172360716998279227991014408, 5.74767439564106201635358380998, 7.18222877076189872125708170246, 9.863472261309497591897124211760, 10.90862461343976034762190192532, 11.82289643901955104567121587191, 13.18255927379790990904826174088, 14.57527884436058790946308844087, 15.78349242099107366674311121800, 16.95837041953224011847663707066