| L(s) = 1 | + (−0.741 − 1.20i)2-s + (−1.99 − 2.37i)3-s + (−0.901 + 1.78i)4-s + (−1.81 + 0.844i)5-s + (−1.38 + 4.15i)6-s + (−0.0212 + 0.0584i)7-s + (2.81 − 0.236i)8-s + (−1.14 + 6.50i)9-s + (2.35 + 1.55i)10-s + (−2.76 − 1.59i)11-s + (6.03 − 1.41i)12-s + (3.32 + 2.32i)13-s + (0.0861 − 0.0176i)14-s + (5.61 + 2.61i)15-s + (−2.37 − 3.21i)16-s + (0.380 − 0.266i)17-s + ⋯ |
| L(s) = 1 | + (−0.524 − 0.851i)2-s + (−1.15 − 1.37i)3-s + (−0.450 + 0.892i)4-s + (−0.809 + 0.377i)5-s + (−0.564 + 1.69i)6-s + (−0.00804 + 0.0220i)7-s + (0.996 − 0.0837i)8-s + (−0.382 + 2.16i)9-s + (0.745 + 0.491i)10-s + (−0.832 − 0.480i)11-s + (1.74 − 0.408i)12-s + (0.922 + 0.646i)13-s + (0.0230 − 0.00472i)14-s + (1.44 + 0.675i)15-s + (−0.593 − 0.804i)16-s + (0.0923 − 0.0646i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.257908 + 0.0409970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.257908 + 0.0409970i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.741 + 1.20i)T \) |
| 37 | \( 1 + (5.84 + 1.66i)T \) |
| good | 3 | \( 1 + (1.99 + 2.37i)T + (-0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (1.81 - 0.844i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (0.0212 - 0.0584i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.76 + 1.59i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.32 - 2.32i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.380 + 0.266i)T + (5.81 - 15.9i)T^{2} \) |
| 19 | \( 1 + (2.95 + 0.258i)T + (18.7 + 3.29i)T^{2} \) |
| 23 | \( 1 + (-1.78 - 0.477i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.62 + 0.702i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-6.47 - 6.47i)T + 31iT^{2} \) |
| 41 | \( 1 + (8.93 - 1.57i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.66 - 7.66i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.88 - 2.24i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.44 - 12.2i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (10.0 + 4.69i)T + (37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (7.59 - 10.8i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (1.84 - 5.07i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (3.29 + 3.93i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 + (5.48 - 2.55i)T + (50.7 - 60.5i)T^{2} \) |
| 83 | \( 1 + (0.0367 - 0.208i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.07 + 8.74i)T + (-57.2 - 68.1i)T^{2} \) |
| 97 | \( 1 + (0.678 - 2.53i)T + (-84.0 - 48.5i)T^{2} \) |
| 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
−11.83814809313487386326011361114, −10.93673052874633582693740721008, −10.58460802732911758006096540668, −8.770060226872413833628748359252, −7.905197941016076144507661624443, −7.11922520297799926111018298702, −6.08135319154331452097120417991, −4.60465247465230433171236899205, −2.95560230298043099853891013368, −1.33326518351352845422420464087,
0.29392501566451976375729296291, 3.89105584438025674006883729465, 4.80878624302323078898906361931, 5.61001933322523730018756457783, 6.65023932121167955896455924990, 8.043736305169010999849216540594, 8.814721987663818607181827702662, 10.09039634557189973674177717470, 10.47230406772943038163073743825, 11.40633946005455428099939722717
