| L(s) = 1 | + (2.27 − 0.0917i)2-s + (−0.278 − 0.960i)3-s + (3.18 − 0.257i)4-s + (−1.29 + 0.492i)5-s + (−0.721 − 2.16i)6-s + (1.28 − 3.01i)7-s + (2.70 − 0.328i)8-s + (−0.845 + 0.534i)9-s + (−2.91 + 1.24i)10-s + (1.23 − 1.95i)11-s + (−1.13 − 2.98i)12-s + (2.73 − 2.34i)13-s + (2.64 − 6.97i)14-s + (0.833 + 1.10i)15-s + (−0.174 + 0.0283i)16-s + (4.22 + 1.79i)17-s + ⋯ |
| L(s) = 1 | + (1.61 − 0.0649i)2-s + (−0.160 − 0.554i)3-s + (1.59 − 0.128i)4-s + (−0.580 + 0.220i)5-s + (−0.294 − 0.882i)6-s + (0.485 − 1.13i)7-s + (0.957 − 0.116i)8-s + (−0.281 + 0.178i)9-s + (−0.920 + 0.392i)10-s + (0.372 − 0.589i)11-s + (−0.327 − 0.862i)12-s + (0.758 − 0.651i)13-s + (0.707 − 1.86i)14-s + (0.215 + 0.286i)15-s + (−0.0435 + 0.00708i)16-s + (1.02 + 0.436i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.74502 - 1.45117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.74502 - 1.45117i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.278 + 0.960i)T \) |
| 13 | \( 1 + (-2.73 + 2.34i)T \) |
| good | 2 | \( 1 + (-2.27 + 0.0917i)T + (1.99 - 0.160i)T^{2} \) |
| 5 | \( 1 + (1.29 - 0.492i)T + (3.74 - 3.31i)T^{2} \) |
| 7 | \( 1 + (-1.28 + 3.01i)T + (-4.84 - 5.04i)T^{2} \) |
| 11 | \( 1 + (-1.23 + 1.95i)T + (-4.71 - 9.93i)T^{2} \) |
| 17 | \( 1 + (-4.22 - 1.79i)T + (11.7 + 12.2i)T^{2} \) |
| 19 | \( 1 + (1.50 + 0.867i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.00 - 6.93i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0925 - 2.29i)T + (-28.9 + 2.33i)T^{2} \) |
| 31 | \( 1 + (2.88 - 3.25i)T + (-3.73 - 30.7i)T^{2} \) |
| 37 | \( 1 + (11.7 + 2.39i)T + (34.0 + 14.5i)T^{2} \) |
| 41 | \( 1 + (1.82 - 0.529i)T + (34.6 - 21.9i)T^{2} \) |
| 43 | \( 1 + (-0.284 - 1.39i)T + (-39.5 + 16.8i)T^{2} \) |
| 47 | \( 1 + (-8.22 - 5.68i)T + (16.6 + 43.9i)T^{2} \) |
| 53 | \( 1 + (0.592 + 4.87i)T + (-51.4 + 12.6i)T^{2} \) |
| 59 | \( 1 + (-1.33 + 8.20i)T + (-55.9 - 18.6i)T^{2} \) |
| 61 | \( 1 + (-5.05 - 3.79i)T + (16.9 + 58.5i)T^{2} \) |
| 67 | \( 1 + (1.21 - 15.0i)T + (-66.1 - 10.7i)T^{2} \) |
| 71 | \( 1 + (3.62 + 3.48i)T + (2.85 + 70.9i)T^{2} \) |
| 73 | \( 1 + (1.90 + 3.63i)T + (-41.4 + 60.0i)T^{2} \) |
| 79 | \( 1 + (-6.07 + 8.79i)T + (-28.0 - 73.8i)T^{2} \) |
| 83 | \( 1 + (-0.966 - 3.91i)T + (-73.4 + 38.5i)T^{2} \) |
| 89 | \( 1 + (0.715 - 0.412i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.90 + 4.81i)T + (19.4 - 95.0i)T^{2} \) |
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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
−11.12087286682570890106392266396, −10.49018227621090120954189429150, −8.755440290504511066001524382638, −7.60204855268787683991614052986, −7.02852143929687222691858378271, −5.89210275888973812848691637436, −5.12490267648816405977287811935, −3.74254550826190579253167249612, −3.38243694481542469132503750056, −1.38007244676671406125431768881,
2.24075702632724327515703903342, 3.57243497749309748164572752308, 4.41021815803733284999825025860, 5.20876234379645957985293585265, 6.03927451165830593094073082427, 7.04798276986233566919623275523, 8.421189912145727104014098676656, 9.161075598400241043075761027438, 10.50415645753199362718069856547, 11.50354067163291545066750686707
