L(s) = 1 | + (0.169 − 0.294i)2-s + (0.5 − 0.866i)3-s + (0.942 + 1.63i)4-s + 5-s + (−0.169 − 0.294i)6-s + (−0.330 − 0.571i)7-s + 1.32·8-s + (−0.499 − 0.866i)9-s + (0.169 − 0.294i)10-s + (−0.339 + 0.588i)11-s + 1.88·12-s + (1.93 + 3.04i)13-s − 0.224·14-s + (0.5 − 0.866i)15-s + (−1.66 + 2.87i)16-s + (−3.71 − 6.43i)17-s + ⋯ |
L(s) = 1 | + (0.120 − 0.208i)2-s + (0.288 − 0.499i)3-s + (0.471 + 0.816i)4-s + 0.447·5-s + (−0.0693 − 0.120i)6-s + (−0.124 − 0.216i)7-s + 0.466·8-s + (−0.166 − 0.288i)9-s + (0.0537 − 0.0930i)10-s + (−0.102 + 0.177i)11-s + 0.544·12-s + (0.535 + 0.844i)13-s − 0.0599·14-s + (0.129 − 0.223i)15-s + (−0.415 + 0.718i)16-s + (−0.900 − 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.54910 - 0.175928i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.54910 - 0.175928i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + (-1.93 - 3.04i)T \) |
good | 2 | \( 1 + (-0.169 + 0.294i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (0.330 + 0.571i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.339 - 0.588i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.71 + 6.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0577 + 0.100i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.88 + 6.72i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.77 - 4.80i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 9.97T + 31T^{2} \) |
| 37 | \( 1 + (4.88 - 8.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.11 - 3.65i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.272 - 0.471i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.01T + 47T^{2} \) |
| 53 | \( 1 - 0.679T + 53T^{2} \) |
| 59 | \( 1 + (1.11 + 1.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.10 - 3.64i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.81 + 6.61i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.65 + 6.33i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 8.01T + 73T^{2} \) |
| 79 | \( 1 - 9.97T + 79T^{2} \) |
| 83 | \( 1 - 1.76T + 83T^{2} \) |
| 89 | \( 1 + (6.77 - 11.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.95 + 8.57i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.59173265670982867393079781954, −11.55137654159798210265768017459, −10.79558917303993318283230998717, −9.305537826701512595998351794482, −8.499557256236971950349337098081, −7.10813651664078485401211453362, −6.67094993085870160368038606220, −4.80640621050534447526874289051, −3.29316567301027369179350186713, −2.02846324427282973056080871056,
1.96018599622287071952816927742, 3.69101048014574956963407307250, 5.36961521232894928434742895297, 6.00063344088715188927538875985, 7.35129735677891017706207058365, 8.703585359359659912085758504516, 9.626959948319683461082888104850, 10.69173995609696905041516149530, 11.13000018025538356019998419633, 12.79947765340061565335161897300