| L(s) = 1 | + (−3.50 − 4.04i)2-s + (−2.93 + 20.3i)4-s + (−6.92 + 15.1i)5-s + (7.38 + 2.16i)7-s + (56.7 − 36.4i)8-s + (85.5 − 25.1i)10-s + (12.2 − 14.0i)11-s + (−28.3 + 8.33i)13-s + (−17.0 − 37.4i)14-s + (−187. − 55.1i)16-s + (6.29 + 43.7i)17-s + (−8.21 + 57.1i)19-s + (−289. − 185. i)20-s − 99.7·22-s + (−104. − 34.4i)23-s + ⋯ |
| L(s) = 1 | + (−1.23 − 1.42i)2-s + (−0.366 + 2.54i)4-s + (−0.619 + 1.35i)5-s + (0.398 + 0.117i)7-s + (2.50 − 1.61i)8-s + (2.70 − 0.794i)10-s + (0.334 − 0.386i)11-s + (−0.605 + 0.177i)13-s + (−0.326 − 0.714i)14-s + (−2.93 − 0.861i)16-s + (0.0898 + 0.624i)17-s + (−0.0992 + 0.690i)19-s + (−3.23 − 2.07i)20-s − 0.966·22-s + (−0.950 − 0.312i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0276639 + 0.0719693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0276639 + 0.0719693i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (104. + 34.4i)T \) |
| good | 2 | \( 1 + (3.50 + 4.04i)T + (-1.13 + 7.91i)T^{2} \) |
| 5 | \( 1 + (6.92 - 15.1i)T + (-81.8 - 94.4i)T^{2} \) |
| 7 | \( 1 + (-7.38 - 2.16i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (-12.2 + 14.0i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (28.3 - 8.33i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-6.29 - 43.7i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (8.21 - 57.1i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (18.6 + 129. i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-128. + 82.3i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (35.6 + 78.0i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (138. - 304. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (168. + 108. i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 + 126.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (728. + 213. i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-711. + 208. i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (221. - 142. i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-66.9 - 77.2i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-180. - 208. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-86.8 + 604. i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (954. - 280. i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (154. + 337. i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (630. + 404. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (153. - 337. i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
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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
−11.67458290609947745217305349594, −11.41496982077188168468637333212, −10.33388080078151870040907906359, −9.788006466618343428187571200685, −8.331573825276684732238371037328, −7.80211702607064052162929331906, −6.54631102390687684966161182081, −4.10640742398448414161775313504, −3.11438076351777445110172133402, −1.90678317549023399475528270530,
0.05325306307493854305127514567, 1.35866669687734079520836144813, 4.60596088854011862218316741210, 5.28767727640837699796526036694, 6.75907649187586154259464288596, 7.71055956441149526105372164038, 8.438973679937471268843063569636, 9.239050448441241471845193090037, 10.07536013046193475458147762736, 11.42560965891221555643452577907
