| L(s) = 1 | + (0.692 + 1.23i)2-s + (1.24 + 0.436i)3-s + (−1.03 + 1.70i)4-s + (1.45 + 0.332i)5-s + (0.326 + 1.83i)6-s + (−2.16 − 4.49i)7-s + (−2.82 − 0.0977i)8-s + (−0.982 − 0.783i)9-s + (0.599 + 2.02i)10-s + (−0.429 + 3.81i)11-s + (−2.04 + 1.67i)12-s + (0.621 − 0.495i)13-s + (4.03 − 5.78i)14-s + (1.67 + 1.05i)15-s + (−1.83 − 3.55i)16-s + (2.22 − 2.22i)17-s + ⋯ |
| L(s) = 1 | + (0.489 + 0.871i)2-s + (0.719 + 0.251i)3-s + (−0.519 + 0.854i)4-s + (0.651 + 0.148i)5-s + (0.133 + 0.750i)6-s + (−0.817 − 1.69i)7-s + (−0.999 − 0.0345i)8-s + (−0.327 − 0.261i)9-s + (0.189 + 0.641i)10-s + (−0.129 + 1.15i)11-s + (−0.589 + 0.483i)12-s + (0.172 − 0.137i)13-s + (1.07 − 1.54i)14-s + (0.431 + 0.271i)15-s + (−0.459 − 0.888i)16-s + (0.539 − 0.539i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22050 + 0.843639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22050 + 0.843639i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.692 - 1.23i)T \) |
| 29 | \( 1 + (-3.73 - 3.87i)T \) |
| good | 3 | \( 1 + (-1.24 - 0.436i)T + (2.34 + 1.87i)T^{2} \) |
| 5 | \( 1 + (-1.45 - 0.332i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (2.16 + 4.49i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (0.429 - 3.81i)T + (-10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (-0.621 + 0.495i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.22 + 2.22i)T - 17iT^{2} \) |
| 19 | \( 1 + (-1.65 - 4.73i)T + (-14.8 + 11.8i)T^{2} \) |
| 23 | \( 1 + (3.02 - 0.689i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (-4.84 + 3.04i)T + (13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (2.57 - 0.290i)T + (36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (-0.977 - 0.977i)T + 41iT^{2} \) |
| 43 | \( 1 + (5.24 - 8.34i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (-7.11 - 0.801i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (-1.32 + 5.81i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + 0.973iT - 59T^{2} \) |
| 61 | \( 1 + (1.52 - 4.36i)T + (-47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (-2.79 + 3.50i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (4.75 + 5.96i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (5.74 - 9.13i)T + (-31.6 - 65.7i)T^{2} \) |
| 79 | \( 1 + (4.41 - 0.497i)T + (77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (-0.599 + 1.24i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-1.59 - 2.53i)T + (-38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (3.39 + 9.69i)T + (-75.8 + 60.4i)T^{2} \) |
| 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
−13.98765509802209413760384164138, −13.19611175784803840827600279897, −12.06296864131134483805583658034, −10.05027440171743928457603548617, −9.681723330067362891148096168870, −8.076848719814156722156753544899, −7.13023480001009534435650766378, −6.02951505173502542579066169654, −4.31427014652972648441493509666, −3.20996008838880052470119095016,
2.30370060487597693009458746707, 3.21433730637152222491773817751, 5.43653441154958551196914400794, 6.12441196994535618598185779472, 8.522744229483422385362351533482, 9.047162522523335776676872532512, 10.15659803694074301135624957269, 11.54024506440635517885054548740, 12.38430872214337552420944744485, 13.51197115451769494394662676193
