L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.730 + 1.26i)3-s + (−0.499 + 0.866i)4-s + (−2.18 − 0.484i)5-s + 1.46·6-s + (−1.48 − 2.18i)7-s + 0.999·8-s + (0.433 + 0.751i)9-s + (0.672 + 2.13i)10-s + (−2.99 − 1.41i)11-s + (−0.730 − 1.26i)12-s − 4.41i·13-s + (−1.15 + 2.38i)14-s + (2.20 − 2.40i)15-s + (−0.5 − 0.866i)16-s + (6.76 + 3.90i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.421 + 0.730i)3-s + (−0.249 + 0.433i)4-s + (−0.976 − 0.216i)5-s + 0.596·6-s + (−0.562 − 0.827i)7-s + 0.353·8-s + (0.144 + 0.250i)9-s + (0.212 + 0.674i)10-s + (−0.904 − 0.426i)11-s + (−0.210 − 0.365i)12-s − 1.22i·13-s + (−0.307 + 0.636i)14-s + (0.569 − 0.621i)15-s + (−0.125 − 0.216i)16-s + (1.64 + 0.946i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.627795 + 0.190131i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.627795 + 0.190131i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (2.18 + 0.484i)T \) |
| 7 | \( 1 + (1.48 + 2.18i)T \) |
| 11 | \( 1 + (2.99 + 1.41i)T \) |
good | 3 | \( 1 + (0.730 - 1.26i)T + (-1.5 - 2.59i)T^{2} \) |
| 13 | \( 1 + 4.41iT - 13T^{2} \) |
| 17 | \( 1 + (-6.76 - 3.90i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.986 - 1.70i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.107 - 0.0621i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9.04iT - 29T^{2} \) |
| 31 | \( 1 + (-2.78 - 1.60i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (9.23 - 5.33i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.22T + 41T^{2} \) |
| 43 | \( 1 - 4.97T + 43T^{2} \) |
| 47 | \( 1 + (1.30 + 2.26i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.19 - 4.73i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.46 - 4.88i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.840 - 1.45i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.21 + 1.27i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + (8.48 + 4.89i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.18 + 3.57i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.51iT - 83T^{2} \) |
| 89 | \( 1 + (0.291 - 0.168i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−10.43138196624122558362609111938, −10.06077444791333452892045667152, −8.680052364693747373882126843253, −7.87419525222568193846909181053, −7.32420015149040541857594974032, −5.66947884011422069642807626216, −4.88398492575113156419535063844, −3.67721525384790391260261645148, −3.21599977189500025996364170342, −0.968854319859111033608555284599,
0.53420471036138909733241331488, 2.40206522149219935647154221370, 3.82231852704746812625792705901, 5.11169848849708742085639668319, 6.01593433687087317137094979969, 6.97995383190091261014713148789, 7.43430707101300486740002944364, 8.308782014521568132454932203214, 9.398502707676113504385870932068, 9.965805634351841119515368500828