| L(s) = 1 | + (−1.19 − 2.31i)2-s + (1.07 − 1.35i)3-s + (−2.77 + 3.89i)4-s + (−0.904 − 3.72i)5-s + (−4.42 − 0.866i)6-s + (2.50 − 0.866i)7-s + (7.17 + 1.03i)8-s + (−0.690 − 2.91i)9-s + (−7.55 + 6.54i)10-s + (0.0853 + 1.79i)11-s + (2.31 + 7.95i)12-s + (−1.08 + 3.13i)13-s + (−4.99 − 4.76i)14-s + (−6.03 − 2.77i)15-s + (−3.04 − 8.79i)16-s + (−0.919 + 2.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.843 − 1.63i)2-s + (0.620 − 0.784i)3-s + (−1.38 + 1.94i)4-s + (−0.404 − 1.66i)5-s + (−1.80 − 0.353i)6-s + (0.946 − 0.327i)7-s + (2.53 + 0.364i)8-s + (−0.230 − 0.973i)9-s + (−2.38 + 2.06i)10-s + (0.0257 + 0.540i)11-s + (0.667 + 2.29i)12-s + (−0.300 + 0.868i)13-s + (−1.33 − 1.27i)14-s + (−1.55 − 0.717i)15-s + (−0.760 − 2.19i)16-s + (−0.222 + 0.488i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.201145 + 0.846387i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.201145 + 0.846387i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.07 + 1.35i)T \) |
| 23 | \( 1 + (-0.537 - 4.76i)T \) |
| good | 2 | \( 1 + (1.19 + 2.31i)T + (-1.16 + 1.62i)T^{2} \) |
| 5 | \( 1 + (0.904 + 3.72i)T + (-4.44 + 2.29i)T^{2} \) |
| 7 | \( 1 + (-2.50 + 0.866i)T + (5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (-0.0853 - 1.79i)T + (-10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (1.08 - 3.13i)T + (-10.2 - 8.03i)T^{2} \) |
| 17 | \( 1 + (0.919 - 2.01i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-5.33 + 2.43i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.362 + 0.258i)T + (9.48 - 27.4i)T^{2} \) |
| 31 | \( 1 + (-3.86 + 1.54i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (2.15 + 7.35i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.19 + 0.532i)T + (36.4 - 18.7i)T^{2} \) |
| 43 | \( 1 + (-0.297 + 0.743i)T + (-31.1 - 29.6i)T^{2} \) |
| 47 | \( 1 + (-2.64 - 1.52i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.12 + 1.30i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (10.9 + 3.79i)T + (46.3 + 36.4i)T^{2} \) |
| 61 | \( 1 + (-2.64 + 3.35i)T + (-14.3 - 59.2i)T^{2} \) |
| 67 | \( 1 + (2.17 + 0.103i)T + (66.6 + 6.36i)T^{2} \) |
| 71 | \( 1 + (-8.51 + 13.2i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.67 - 3.67i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.836 - 4.34i)T + (-73.3 + 29.3i)T^{2} \) |
| 83 | \( 1 + (3.00 - 12.3i)T + (-73.7 - 38.0i)T^{2} \) |
| 89 | \( 1 + (-0.218 - 1.51i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-9.85 - 10.3i)T + (-4.61 + 96.8i)T^{2} \) |
| 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
−11.92858551025827545337255041540, −11.16075030152060175351645078775, −9.503583271086039719810767069032, −9.116645998473816549355459835833, −8.117190661721144028834542452641, −7.50764679551697935774137123199, −4.85223778239428007171220712959, −3.83412920604553631581563368716, −1.98862458025171330207665647630, −1.03958583122556221444488694000,
3.01381035098886091822375919369, 4.78920799257180737234116332316, 5.91862789514961560613544497021, 7.19716451375275179598358589190, 7.919855906607124733991630891401, 8.635314843217405701558367978683, 9.922948408142436338857401506839, 10.52089383856267414746767239398, 11.51708214470174177447541078515, 13.86839809535656477544041813665
