L(s) = 1 | + (−1.15e5 + 1.15e5i)2-s + (1.39e8 + 1.39e8i)3-s − 9.46e9i·4-s + (−7.62e11 + 3.18e10i)5-s − 3.21e13·6-s + (−2.97e14 + 2.97e14i)7-s + (−8.90e14 − 8.90e14i)8-s + 2.21e16i·9-s + (8.43e16 − 9.16e16i)10-s − 1.99e17·11-s + (1.31e18 − 1.31e18i)12-s + (3.71e18 + 3.71e18i)13-s − 6.87e19i·14-s + (−1.10e20 − 1.01e20i)15-s + 3.68e20·16-s + (−4.39e20 + 4.39e20i)17-s + ⋯ |
L(s) = 1 | + (−0.880 + 0.880i)2-s + (1.07 + 1.07i)3-s − 0.551i·4-s + (−0.999 + 0.0417i)5-s − 1.90·6-s + (−1.28 + 1.28i)7-s + (−0.395 − 0.395i)8-s + 1.32i·9-s + (0.843 − 0.916i)10-s − 0.394·11-s + (0.594 − 0.594i)12-s + (0.429 + 0.429i)13-s − 2.25i·14-s + (−1.12 − 1.03i)15-s + 1.24·16-s + (−0.531 + 0.531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{35}{2})\) |
\(\approx\) |
\(0.4103363169\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4103363169\) |
\(L(18)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (7.62e11 - 3.18e10i)T \) |
good | 2 | \( 1 + (1.15e5 - 1.15e5i)T - 1.71e10iT^{2} \) |
| 3 | \( 1 + (-1.39e8 - 1.39e8i)T + 1.66e16iT^{2} \) |
| 7 | \( 1 + (2.97e14 - 2.97e14i)T - 5.41e28iT^{2} \) |
| 11 | \( 1 + 1.99e17T + 2.55e35T^{2} \) |
| 13 | \( 1 + (-3.71e18 - 3.71e18i)T + 7.48e37iT^{2} \) |
| 17 | \( 1 + (4.39e20 - 4.39e20i)T - 6.84e41iT^{2} \) |
| 19 | \( 1 - 7.08e21iT - 3.00e43T^{2} \) |
| 23 | \( 1 + (1.16e23 + 1.16e23i)T + 1.98e46iT^{2} \) |
| 29 | \( 1 - 7.43e23iT - 5.26e49T^{2} \) |
| 31 | \( 1 - 1.56e25T + 5.08e50T^{2} \) |
| 37 | \( 1 + (4.19e26 - 4.19e26i)T - 2.08e53iT^{2} \) |
| 41 | \( 1 - 4.69e27T + 6.83e54T^{2} \) |
| 43 | \( 1 + (-1.38e27 - 1.38e27i)T + 3.45e55iT^{2} \) |
| 47 | \( 1 + (1.72e28 - 1.72e28i)T - 7.10e56iT^{2} \) |
| 53 | \( 1 + (6.65e28 + 6.65e28i)T + 4.22e58iT^{2} \) |
| 59 | \( 1 - 1.60e30iT - 1.61e60T^{2} \) |
| 61 | \( 1 - 1.31e30T + 5.02e60T^{2} \) |
| 67 | \( 1 + (2.97e30 - 2.97e30i)T - 1.22e62iT^{2} \) |
| 71 | \( 1 - 3.43e31T + 8.76e62T^{2} \) |
| 73 | \( 1 + (-9.17e30 - 9.17e30i)T + 2.25e63iT^{2} \) |
| 79 | \( 1 + 8.66e31iT - 3.30e64T^{2} \) |
| 83 | \( 1 + (2.22e32 + 2.22e32i)T + 1.77e65iT^{2} \) |
| 89 | \( 1 + 1.58e33iT - 1.90e66T^{2} \) |
| 97 | \( 1 + (1.71e33 - 1.71e33i)T - 3.55e67iT^{2} \) |
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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
−16.16957928991885401341090737752, −15.83315459501577042436791523984, −14.79886006916363941704019359641, −12.42726069074927818761645713458, −10.03403382600438894226140513828, −8.880345334230463821334575219740, −8.151894192346983256241516592173, −6.26969464419431786664059744160, −3.92765634075958360119298736362, −2.84711163108888968551928360160,
0.18553380279715312871192419732, 0.887747576185092956052578850381, 2.57773120869037436546279015836, 3.58740342600202345615821837073, 6.95267044393955006444895099502, 8.030403356672993407395917875709, 9.390069307505795895710641303079, 10.95116258577712668707433319456, 12.64766866657173030335065171802, 13.76605440451046511856754028206