| L(s) = 1 | + (−0.936 − 1.05i)2-s + (0.164 + 0.0216i)3-s + (−0.246 + 1.98i)4-s + (−0.755 + 0.0994i)5-s + (−0.130 − 0.194i)6-s + (−0.876 − 2.49i)7-s + (2.33 − 1.59i)8-s + (−2.87 − 0.769i)9-s + (0.812 + 0.707i)10-s + (2.95 − 2.26i)11-s + (−0.0835 + 0.320i)12-s + (−1.74 − 4.22i)13-s + (−1.82 + 3.26i)14-s − 0.126·15-s + (−3.87 − 0.980i)16-s + (3.44 − 5.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.662 − 0.749i)2-s + (0.0949 + 0.0124i)3-s + (−0.123 + 0.992i)4-s + (−0.337 + 0.0444i)5-s + (−0.0534 − 0.0793i)6-s + (−0.331 − 0.943i)7-s + (0.825 − 0.564i)8-s + (−0.957 − 0.256i)9-s + (0.257 + 0.223i)10-s + (0.890 − 0.683i)11-s + (−0.0241 + 0.0926i)12-s + (−0.485 − 1.17i)13-s + (−0.487 + 0.872i)14-s − 0.0326·15-s + (−0.969 − 0.245i)16-s + (0.835 − 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.283004 - 0.605110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.283004 - 0.605110i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.936 + 1.05i)T \) |
| 7 | \( 1 + (0.876 + 2.49i)T \) |
| good | 3 | \( 1 + (-0.164 - 0.0216i)T + (2.89 + 0.776i)T^{2} \) |
| 5 | \( 1 + (0.755 - 0.0994i)T + (4.82 - 1.29i)T^{2} \) |
| 11 | \( 1 + (-2.95 + 2.26i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.74 + 4.22i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (-3.44 + 5.96i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.01 - 0.781i)T + (4.91 + 18.3i)T^{2} \) |
| 23 | \( 1 + (1.63 - 6.09i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (0.478 + 1.15i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (1.64 - 2.84i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.24 + 9.42i)T + (-35.7 + 9.57i)T^{2} \) |
| 41 | \( 1 + (-7.52 - 7.52i)T + 41iT^{2} \) |
| 43 | \( 1 + (-5.35 - 2.21i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-4.87 + 2.81i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.62 - 2.11i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (-10.5 + 8.10i)T + (15.2 - 56.9i)T^{2} \) |
| 61 | \( 1 + (8.78 + 6.73i)T + (15.7 + 58.9i)T^{2} \) |
| 67 | \( 1 + (1.21 - 9.22i)T + (-64.7 - 17.3i)T^{2} \) |
| 71 | \( 1 + (5.31 - 5.31i)T - 71iT^{2} \) |
| 73 | \( 1 + (-4.98 + 1.33i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.13 - 5.43i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.31 + 2.61i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (11.2 + 3.01i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 0.918iT - 97T^{2} \) |
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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
−11.64874796058923750633432179035, −11.07924182042074022973455951962, −9.863288845464347998506242906744, −9.229306665359626266595840533170, −7.917818436166543243989223412943, −7.30846656359993303102810771828, −5.64794847746080762012968221897, −3.81685445874597456673437774469, −3.02749175176592373859627121654, −0.68788916700983352719206661571,
2.13080289339566821547461511022, 4.24327139375381362257674837525, 5.69627135015551798658925189041, 6.50653695014370182747641951577, 7.73936164704045488116909219329, 8.723818971041789068270487004215, 9.351749412971755442986091541334, 10.44699213700450941630926751094, 11.71609686570200275407748577435, 12.33724059682055262451188810004
