L(s) = 1 | + (1.66 + 0.447i)3-s + (2.21 − 0.298i)5-s + (0.900 − 2.48i)7-s + (−0.0138 − 0.00800i)9-s + (−1.10 − 1.91i)11-s + (−1.91 + 1.91i)13-s + (3.83 + 0.491i)15-s + (−1.12 + 4.19i)17-s + (−1.80 + 3.12i)19-s + (2.61 − 3.74i)21-s + (0.375 − 0.100i)23-s + (4.82 − 1.32i)25-s + (−3.68 − 3.68i)27-s + 6.62i·29-s + (0.897 − 0.518i)31-s + ⋯ |
L(s) = 1 | + (0.963 + 0.258i)3-s + (0.991 − 0.133i)5-s + (0.340 − 0.940i)7-s + (−0.00462 − 0.00266i)9-s + (−0.333 − 0.578i)11-s + (−0.530 + 0.530i)13-s + (0.989 + 0.127i)15-s + (−0.272 + 1.01i)17-s + (−0.413 + 0.716i)19-s + (0.570 − 0.817i)21-s + (0.0783 − 0.0209i)23-s + (0.964 − 0.264i)25-s + (−0.708 − 0.708i)27-s + 1.22i·29-s + (0.161 − 0.0930i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.87783 - 0.101703i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87783 - 0.101703i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.21 + 0.298i)T \) |
| 7 | \( 1 + (-0.900 + 2.48i)T \) |
good | 3 | \( 1 + (-1.66 - 0.447i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (1.10 + 1.91i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.91 - 1.91i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.12 - 4.19i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.80 - 3.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.375 + 0.100i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 6.62iT - 29T^{2} \) |
| 31 | \( 1 + (-0.897 + 0.518i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.84 - 6.88i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 4.03iT - 41T^{2} \) |
| 43 | \( 1 + (8.37 + 8.37i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.52 + 1.48i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.62 + 6.07i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.71 - 6.42i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.84 - 4.52i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.0 + 2.70i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 9.39T + 71T^{2} \) |
| 73 | \( 1 + (13.7 + 3.69i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.38 - 3.11i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.77 + 7.77i)T - 83iT^{2} \) |
| 89 | \( 1 + (-5.20 + 9.01i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.1 + 12.1i)T + 97iT^{2} \) |
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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
−11.83482341641431181223491495753, −10.53720697084875460384763363954, −10.03972804343650294452008277998, −8.859498322762244304542250395269, −8.280558693854108670159107220844, −6.98613542246045609662838373934, −5.82785935844782355012351436663, −4.47912983954489465897038621581, −3.24836573776926843298366189306, −1.81125366035995522035335737212,
2.21591204776973086227417970130, 2.75492855930375546857233662021, 4.83606377335671619185981552334, 5.78838524164624799487554021059, 7.12088696440841854524980363840, 8.118213851526890409894559831266, 9.095154615055607764019390057697, 9.679170060317167321458416326032, 10.87875369585194682344880682976, 11.98688242485529197172928701290