| L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.198 − 0.171i)3-s + (0.841 − 0.540i)4-s + (0.382 + 2.66i)5-s + (−0.141 + 0.220i)6-s + (2.47 − 0.927i)7-s + (−0.654 + 0.755i)8-s + (−0.417 + 2.90i)9-s + (−1.11 − 2.44i)10-s + (1.12 − 3.81i)11-s + (0.0739 − 0.251i)12-s + (−4.25 + 1.94i)13-s + (−2.11 + 1.58i)14-s + (0.533 + 0.462i)15-s + (0.415 − 0.909i)16-s + (5.41 + 3.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.114 − 0.0992i)3-s + (0.420 − 0.270i)4-s + (0.171 + 1.19i)5-s + (−0.0579 + 0.0901i)6-s + (0.936 − 0.350i)7-s + (−0.231 + 0.267i)8-s + (−0.139 + 0.967i)9-s + (−0.353 − 0.773i)10-s + (0.337 − 1.15i)11-s + (0.0213 − 0.0727i)12-s + (−1.17 + 0.538i)13-s + (−0.565 + 0.424i)14-s + (0.137 + 0.119i)15-s + (0.103 − 0.227i)16-s + (1.31 + 0.843i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.918946 + 0.574418i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.918946 + 0.574418i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (-2.47 + 0.927i)T \) |
| 23 | \( 1 + (-4.50 + 1.63i)T \) |
| good | 3 | \( 1 + (-0.198 + 0.171i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.382 - 2.66i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (-1.12 + 3.81i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (4.25 - 1.94i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.41 - 3.47i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (5.37 - 3.45i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-8.32 - 5.34i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (0.278 + 0.241i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (5.74 + 0.825i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.832 - 0.119i)T + (39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.45 + 1.26i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 1.02iT - 47T^{2} \) |
| 53 | \( 1 + (8.17 + 3.73i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (1.80 - 0.824i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-4.61 + 5.32i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (1.06 + 3.61i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-1.23 + 0.362i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (7.80 + 12.1i)T + (-30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.00 + 0.460i)T + (51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.11 + 14.7i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-2.64 - 3.05i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (2.39 + 16.6i)T + (-93.0 + 27.3i)T^{2} \) |
| 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
−11.44184434331862727455908122327, −10.50406781396728972453950508981, −10.39506827145260865836330897449, −8.736745877894090127590430354844, −8.012152153652789140102355346802, −7.14069513129603457076771952453, −6.17969596727272076491827569805, −4.89546967556411869061465098036, −3.15912198421828077936615867662, −1.80865699086160211479880628972,
1.06041770759212088589301448318, 2.61505970859597953187226223063, 4.50172645118542330531625534392, 5.27193479265814549061857486193, 6.84871653597159898250150556106, 7.88604156040926092669330083387, 8.817109306263061114928919203267, 9.461797740013420582168757820924, 10.25172383189848283881168642717, 11.69120986535235773020664137878
