| L(s) = 1 | + (0.415 + 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.0867 + 0.603i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (0.142 − 0.989i)10-s + (2.32 − 5.09i)11-s + (0.415 − 0.909i)12-s + (0.602 − 4.19i)13-s + (−0.512 + 0.329i)14-s + (0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (2.45 + 2.83i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (−0.376 − 0.241i)5-s + (−0.267 − 0.308i)6-s + (0.0327 + 0.227i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.0450 − 0.313i)10-s + (0.701 − 1.53i)11-s + (0.119 − 0.262i)12-s + (0.167 − 1.16i)13-s + (−0.136 + 0.0880i)14-s + (0.247 + 0.0727i)15-s + (−0.0355 − 0.247i)16-s + (0.596 + 0.688i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32086 + 0.103291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32086 + 0.103291i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (1.17 - 4.64i)T \) |
| good | 7 | \( 1 + (-0.0867 - 0.603i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-2.32 + 5.09i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.602 + 4.19i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.45 - 2.83i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.15 + 1.33i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (0.762 + 0.880i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-8.18 - 2.40i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-1.99 + 1.28i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (3.48 + 2.24i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-8.12 + 2.38i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 + (1.78 + 12.4i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.70 + 11.8i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (1.61 + 0.473i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-4.60 - 10.0i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (5.15 + 11.2i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (1.54 - 1.78i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.25 - 8.74i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (13.0 - 8.37i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-3.57 + 1.04i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (2.76 + 1.77i)T + (40.2 + 88.2i)T^{2} \) |
| 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
−10.58528288591754601965230936670, −9.546770497951770454836566679730, −8.491493742290786635866781339343, −7.994123096486667464783610655050, −6.80122252551499010476661391083, −5.80064727544585831337295927829, −5.39301752912898561667283727116, −4.01390903849246576835459177911, −3.21241850870509334642916699065, −0.830002667289150858411505779126,
1.27993566496376174169907844756, 2.61196855749864473853930794896, 4.22156149985769499563465381566, 4.53943814015489859759505025933, 5.98178683454243473416577425969, 6.89152514994754894553167336459, 7.62088645888425022344396724897, 8.994804563384384444202370863304, 9.814902462257633740101227055146, 10.49113672914509283673264723343
