| 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
−11.50354067163291545066750686707, −10.50415645753199362718069856547, −9.161075598400241043075761027438, −8.421189912145727104014098676656, −7.04798276986233566919623275523, −6.03927451165830593094073082427, −5.20876234379645957985293585265, −4.41021815803733284999825025860, −3.57243497749309748164572752308, −2.24075702632724327515703903342,
1.38007244676671406125431768881, 3.38243694481542469132503750056, 3.74254550826190579253167249612, 5.12490267648816405977287811935, 5.89210275888973812848691637436, 7.02852143929687222691858378271, 7.60204855268787683991614052986, 8.755440290504511066001524382638, 10.49018227621090120954189429150, 11.12087286682570890106392266396
