L(s) = 1 | + (1.29 − 1.82i)5-s + (0.306 − 0.306i)7-s − 4.06·11-s + (−0.625 + 0.625i)13-s + (−3.57 − 3.57i)17-s − 6.82·19-s + (−1.58 + 1.58i)23-s + (−1.65 − 4.71i)25-s − 8.50i·29-s + 2.56·31-s + (−0.163 − 0.955i)35-s + (1.69 + 1.69i)37-s + 5.19i·41-s + (−4.18 + 4.18i)43-s + (4.58 + 4.58i)47-s + ⋯ |
L(s) = 1 | + (0.578 − 0.815i)5-s + (0.115 − 0.115i)7-s − 1.22·11-s + (−0.173 + 0.173i)13-s + (−0.867 − 0.867i)17-s − 1.56·19-s + (−0.331 + 0.331i)23-s + (−0.331 − 0.943i)25-s − 1.58i·29-s + 0.461·31-s + (−0.0275 − 0.161i)35-s + (0.277 + 0.277i)37-s + 0.811i·41-s + (−0.638 + 0.638i)43-s + (0.668 + 0.668i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7006799370\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7006799370\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.29 + 1.82i)T \) |
good | 7 | \( 1 + (-0.306 + 0.306i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 + (0.625 - 0.625i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.57 + 3.57i)T + 17iT^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + (1.58 - 1.58i)T - 23iT^{2} \) |
| 29 | \( 1 + 8.50iT - 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 + (-1.69 - 1.69i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + (4.18 - 4.18i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.58 - 4.58i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.41 + 7.41i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.79iT - 59T^{2} \) |
| 61 | \( 1 - 6.08iT - 61T^{2} \) |
| 67 | \( 1 + (6.18 + 6.18i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.7iT - 71T^{2} \) |
| 73 | \( 1 + (-3.05 - 3.05i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.56iT - 79T^{2} \) |
| 83 | \( 1 + (5.13 + 5.13i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.88T + 89T^{2} \) |
| 97 | \( 1 + (-1.75 + 1.75i)T - 97iT^{2} \) |
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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
−9.283558061648344031343291256868, −8.271317420199574887823209334377, −7.83682826635907260197916255636, −6.57047668423243535451906148483, −5.90295582356171304653219569243, −4.79587278258603415019541028049, −4.41147417696126865400665146610, −2.75741954908906482184557058401, −1.91196774373759151715084023632, −0.25070698643963224581451642223,
1.96530618670430467857526885000, 2.66020273459181548705949306165, 3.85721616522791320819897631822, 4.96320980695050115624298886757, 5.84554034958697384863639607124, 6.60451260108518309250538615266, 7.37384234638330908902551219436, 8.385667105011138817696142708177, 8.948750048593418412408630803194, 10.23042434036282709646269017306