| L(s) = 1 | + (0.775 − 0.195i)2-s + (0.868 + 1.15i)3-s + (−1.19 + 0.643i)4-s + (−1.15 + 1.34i)5-s + (0.898 + 0.722i)6-s + (4.84 − 1.54i)7-s + (−1.98 + 1.81i)8-s + (0.252 − 0.886i)9-s + (−0.632 + 1.27i)10-s + (−2.07 − 2.13i)11-s + (−1.78 − 0.819i)12-s + (−3.35 − 3.06i)13-s + (3.46 − 2.14i)14-s + (−2.55 − 0.157i)15-s + (0.315 − 0.475i)16-s + (−1.17 + 0.946i)17-s + ⋯ |
| L(s) = 1 | + (0.548 − 0.137i)2-s + (0.501 + 0.664i)3-s + (−0.599 + 0.321i)4-s + (−0.516 + 0.602i)5-s + (0.366 + 0.295i)6-s + (1.83 − 0.582i)7-s + (−0.702 + 0.640i)8-s + (0.0841 − 0.295i)9-s + (−0.200 + 0.401i)10-s + (−0.624 − 0.644i)11-s + (−0.514 − 0.236i)12-s + (−0.931 − 0.848i)13-s + (0.924 − 0.572i)14-s + (−0.659 − 0.0406i)15-s + (0.0788 − 0.118i)16-s + (−0.285 + 0.229i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26391 + 0.378181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26391 + 0.378181i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (3.54 - 9.51i)T \) |
| good | 2 | \( 1 + (-0.775 + 0.195i)T + (1.76 - 0.946i)T^{2} \) |
| 3 | \( 1 + (-0.868 - 1.15i)T + (-0.820 + 2.88i)T^{2} \) |
| 5 | \( 1 + (1.15 - 1.34i)T + (-0.766 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-4.84 + 1.54i)T + (5.71 - 4.04i)T^{2} \) |
| 11 | \( 1 + (2.07 + 2.13i)T + (-0.338 + 10.9i)T^{2} \) |
| 13 | \( 1 + (3.35 + 3.06i)T + (1.19 + 12.9i)T^{2} \) |
| 17 | \( 1 + (1.17 - 0.946i)T + (3.63 - 16.6i)T^{2} \) |
| 19 | \( 1 + (-0.00273 - 0.000338i)T + (18.4 + 4.63i)T^{2} \) |
| 23 | \( 1 + (-1.69 - 5.95i)T + (-19.5 + 12.1i)T^{2} \) |
| 29 | \( 1 + (0.968 + 2.74i)T + (-22.5 + 18.1i)T^{2} \) |
| 31 | \( 1 + (3.29 + 6.61i)T + (-18.6 + 24.7i)T^{2} \) |
| 37 | \( 1 + (-0.425 - 4.59i)T + (-36.3 + 6.79i)T^{2} \) |
| 41 | \( 1 + (-2.43 - 2.84i)T + (-6.28 + 40.5i)T^{2} \) |
| 43 | \( 1 + (8.47 - 3.89i)T + (27.9 - 32.6i)T^{2} \) |
| 47 | \( 1 + (2.61 - 4.53i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.920 - 2.17i)T + (-36.8 + 38.0i)T^{2} \) |
| 59 | \( 1 + (12.2 + 3.90i)T + (48.1 + 34.0i)T^{2} \) |
| 61 | \( 1 + (-3.43 - 1.33i)T + (45.0 + 41.0i)T^{2} \) |
| 67 | \( 1 + (-0.126 - 0.578i)T + (-60.8 + 28.0i)T^{2} \) |
| 71 | \( 1 + (4.89 - 13.8i)T + (-55.3 - 44.5i)T^{2} \) |
| 73 | \( 1 + (-10.3 + 1.93i)T + (68.0 - 26.3i)T^{2} \) |
| 79 | \( 1 + (-5.89 - 1.10i)T + (73.6 + 28.5i)T^{2} \) |
| 83 | \( 1 + (-0.349 + 1.59i)T + (-75.4 - 34.6i)T^{2} \) |
| 89 | \( 1 + (4.15 - 2.57i)T + (39.6 - 79.6i)T^{2} \) |
| 97 | \( 1 + (6.84 + 5.50i)T + (20.7 + 94.7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.06035917981032922436641594836, −13.09363558301784429381156111853, −11.66352035546117090110523258343, −10.95425049541311943460933412347, −9.638108042144460250365591437079, −8.238642557031213698803985595431, −7.64198965732944899860995987810, −5.29201981591040331792810036000, −4.27798320658063781327755748970, −3.12606055359777621534789375999,
2.02911388671503349813834299477, 4.68731087997561545791363375620, 5.00149647214560043089345727084, 7.15258595520270611675346604025, 8.261776582695266946419728601438, 8.973368117881696977587525653453, 10.65606991058119828958351453986, 12.11039138113088062540284306415, 12.63398572262366699937445818506, 13.91896200148586463591930191031
