| L(s) = 1 | + (−0.161 − 1.53i)2-s + (1.18 + 1.26i)3-s + (−0.382 + 0.0811i)4-s + (−0.0618 + 0.588i)5-s + (1.75 − 2.02i)6-s + (0.483 + 0.536i)7-s + (−0.768 − 2.36i)8-s + (−0.195 + 2.99i)9-s + 0.915·10-s + (−2.56 − 2.10i)11-s + (−0.555 − 0.386i)12-s + (−1.43 − 0.640i)13-s + (0.746 − 0.829i)14-s + (−0.817 + 0.619i)15-s + (−4.22 + 1.88i)16-s + (−3.71 − 2.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.114 − 1.08i)2-s + (0.683 + 0.729i)3-s + (−0.191 + 0.0405i)4-s + (−0.0276 + 0.263i)5-s + (0.715 − 0.826i)6-s + (0.182 + 0.202i)7-s + (−0.271 − 0.836i)8-s + (−0.0650 + 0.997i)9-s + 0.289·10-s + (−0.773 − 0.634i)11-s + (−0.160 − 0.111i)12-s + (−0.399 − 0.177i)13-s + (0.199 − 0.221i)14-s + (−0.211 + 0.159i)15-s + (−1.05 + 0.470i)16-s + (−0.901 − 0.654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08964 - 0.423285i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08964 - 0.423285i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.18 - 1.26i)T \) |
| 11 | \( 1 + (2.56 + 2.10i)T \) |
| good | 2 | \( 1 + (0.161 + 1.53i)T + (-1.95 + 0.415i)T^{2} \) |
| 5 | \( 1 + (0.0618 - 0.588i)T + (-4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (-0.483 - 0.536i)T + (-0.731 + 6.96i)T^{2} \) |
| 13 | \( 1 + (1.43 + 0.640i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (3.71 + 2.69i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.775 - 2.38i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.22 - 3.85i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.66 - 5.18i)T + (-3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (8.33 + 3.71i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.893 + 2.75i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.772 - 0.857i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.10 + 3.64i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.222 - 0.0473i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-4.61 + 3.35i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (6.97 - 1.48i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-4.16 + 1.85i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-4.04 - 7.00i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.95 - 7.23i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.78 + 14.7i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.52 - 14.5i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (6.35 - 2.82i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + (1.48 + 14.1i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−13.65044586440719539146778793424, −12.67184596672927754975132692035, −11.27386696354149792948277604612, −10.71071799651174548648601591419, −9.659634776269834356801065904649, −8.680203346625037862718936760203, −7.21693735846643514523985405844, −5.22333693720605082856483302440, −3.53590511326420557068958493235, −2.43218893121583165715225348474,
2.45454291234060176464553928727, 4.79335113674126833659743805873, 6.43713897751104210038301820905, 7.30418714108398556806497281945, 8.239837734255056209196578898812, 9.155443007365251377687495710893, 10.82867914298960575960403406644, 12.25269434032037523478259693964, 13.14639826282859138342945544218, 14.26817720990760968543449241426
