| L(s) = 1 | − 2-s + 4-s + (1.93 + 1.80i)7-s − 8-s + 3.87i·11-s + 1.60·13-s + (−1.93 − 1.80i)14-s + 16-s − 8.11i·17-s − 2.63i·19-s − 3.87i·22-s − 5.47·23-s − 1.60·26-s + (1.93 + 1.80i)28-s − 5.47i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.732 + 0.681i)7-s − 0.353·8-s + 1.16i·11-s + 0.445·13-s + (−0.517 − 0.481i)14-s + 0.250·16-s − 1.96i·17-s − 0.605i·19-s − 0.825i·22-s − 1.14·23-s − 0.314·26-s + (0.366 + 0.340i)28-s − 1.01i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223682531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223682531\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.93 - 1.80i)T \) |
| good | 11 | \( 1 - 3.87iT - 11T^{2} \) |
| 13 | \( 1 - 1.60T + 13T^{2} \) |
| 17 | \( 1 + 8.11iT - 17T^{2} \) |
| 19 | \( 1 + 2.63iT - 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 + 5.47iT - 29T^{2} \) |
| 31 | \( 1 + 3.73iT - 31T^{2} \) |
| 37 | \( 1 + 4.51iT - 37T^{2} \) |
| 41 | \( 1 + 1.60T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 - 2.26T + 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 - 11.8iT - 61T^{2} \) |
| 67 | \( 1 + 6.90iT - 67T^{2} \) |
| 71 | \( 1 - 2.63iT - 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 - 3.20iT - 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 - 8.68T + 97T^{2} \) |
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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
−8.683930898360714435751768626879, −7.76885446252274662300643666804, −7.32365548601391587273415903066, −6.47509475382714845995061365944, −5.49297620339672565989241376035, −4.84512513366339356179159237586, −3.85565412775405198212443918295, −2.45258268324297001481658316822, −2.03798989673770666836367162478, −0.53098921986875441939843042025,
1.11863467547930060166929756175, 1.81034895927463876159996144861, 3.28935920037945311442685445609, 3.90440265194172653559021790835, 4.98294481008181118096478608761, 6.13011554513155647229932354722, 6.35682395818301436912241857373, 7.61795014016908908532407959328, 8.216002848958300643993939539702, 8.475559110350237719847998299363
