L(s) = 1 | + (−0.342 − 0.939i)2-s + (1.85 + 0.326i)3-s + (−0.766 + 0.642i)4-s + (−0.326 − 1.85i)6-s + (4.38 − 2.53i)7-s + (0.866 + 0.500i)8-s + (0.5 + 0.181i)9-s + (0.705 − 1.22i)11-s + (−1.62 + 0.939i)12-s + (−1.28 + 0.226i)13-s + (−3.87 − 3.25i)14-s + (0.173 − 0.984i)16-s + (−0.817 − 2.24i)17-s − 0.532i·18-s + (−2.23 − 3.74i)19-s + ⋯ |
L(s) = 1 | + (−0.241 − 0.664i)2-s + (1.06 + 0.188i)3-s + (−0.383 + 0.321i)4-s + (−0.133 − 0.755i)6-s + (1.65 − 0.957i)7-s + (0.306 + 0.176i)8-s + (0.166 + 0.0606i)9-s + (0.212 − 0.368i)11-s + (−0.469 + 0.271i)12-s + (−0.356 + 0.0628i)13-s + (−1.03 − 0.869i)14-s + (0.0434 − 0.246i)16-s + (−0.198 − 0.544i)17-s − 0.125i·18-s + (−0.512 − 0.858i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75896 - 1.31350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75896 - 1.31350i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.23 + 3.74i)T \) |
good | 3 | \( 1 + (-1.85 - 0.326i)T + (2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (-4.38 + 2.53i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.705 + 1.22i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.28 - 0.226i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.817 + 2.24i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.96 - 2.34i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.94 - 2.89i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.184 + 0.320i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.82iT - 37T^{2} \) |
| 41 | \( 1 + (0.266 - 1.50i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.487 - 0.581i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (3.49 - 9.59i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (1.07 + 1.28i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.673 + 0.245i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.47 + 6.27i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (0.480 - 1.31i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (4.87 + 4.09i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-4.49 - 0.791i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.389 + 2.20i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-3.45 + 1.99i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.84 - 10.4i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.524 - 1.43i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−9.812730485133027099365419414547, −8.985227949916122533304289464667, −8.301619934487388210611150253203, −7.75042068074347169286441749969, −6.75398556207348582869908376144, −5.00747510256422385714874625165, −4.42130101933174026231558438731, −3.32780852786593095822506171035, −2.33632447480559448969649039650, −1.10722065392265758350151640006,
1.70395939253872750819905285909, 2.52199654723126744159355379671, 4.09275936025498350333747781819, 5.02866381793449584711245870904, 5.89948376382424799009739331183, 7.07428613066698003061384846643, 8.017404044706194784415898083908, 8.426934752434676122393540593014, 8.916017024030480013593393127248, 9.985980004143702752713697721891