| L(s) = 1 | + (1.62 + 0.586i)3-s + (−1.53 − 1.28i)4-s + (2.20 + 0.388i)5-s + (−2.02 + 1.70i)7-s + (2.31 + 1.91i)9-s + (4.46 − 0.787i)11-s + (−1.74 − 2.99i)12-s + (−6.63 + 2.41i)13-s + (3.36 + 1.92i)15-s + (0.694 + 3.93i)16-s + (5.90 + 3.40i)17-s + (−2.87 − 3.42i)20-s + (−4.30 + 1.58i)21-s + (4.69 + 1.71i)25-s + (2.64 + 4.47i)27-s + 5.29·28-s + ⋯ |
| L(s) = 1 | + (0.940 + 0.338i)3-s + (−0.766 − 0.642i)4-s + (0.984 + 0.173i)5-s + (−0.766 + 0.642i)7-s + (0.770 + 0.637i)9-s + (1.34 − 0.237i)11-s + (−0.503 − 0.864i)12-s + (−1.84 + 0.670i)13-s + (0.867 + 0.496i)15-s + (0.173 + 0.984i)16-s + (1.43 + 0.826i)17-s + (−0.642 − 0.766i)20-s + (−0.938 + 0.345i)21-s + (0.939 + 0.342i)25-s + (0.509 + 0.860i)27-s + 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.92718 + 0.777502i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92718 + 0.777502i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.62 - 0.586i)T \) |
| 5 | \( 1 + (-2.20 - 0.388i)T \) |
| 7 | \( 1 + (2.02 - 1.70i)T \) |
| good | 2 | \( 1 + (1.53 + 1.28i)T^{2} \) |
| 11 | \( 1 + (-4.46 + 0.787i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (6.63 - 2.41i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-5.90 - 3.40i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.65 + 4.54i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.27 - 7.47i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.16 - 0.675i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.36 + 14.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.58 + 2.39i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.73 + 12.9i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.71 - 9.72i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.948739639724458729002386477275, −9.237654695003902827633238642411, −9.042312591404464262843975717053, −7.75450301684364284582825709208, −6.58374848872992957545758662147, −5.82598276576132059160036531587, −4.84301181275180108536747299662, −3.83495456068343088805902648797, −2.67550045932323310845874737182, −1.56654368167469072251757109475,
0.991492463747125168300394936407, 2.63603439190761004940586820424, 3.42001446716366840145974399062, 4.47410674308361848154211583676, 5.51053382108762143170656546360, 6.98294965264064774164492297353, 7.26568397324393703616572204863, 8.372814714027634442138095409008, 9.313584306090403579400112998058, 9.711366989883756317098162065441
