L(s) = 1 | + (1.36 + 0.788i)2-s + (−1.67 − 0.432i)3-s + (0.243 + 0.421i)4-s − 5-s + (−1.94 − 1.91i)6-s + (0.569 − 2.58i)7-s − 2.38i·8-s + (2.62 + 1.45i)9-s + (−1.36 − 0.788i)10-s − 4.04i·11-s + (−0.225 − 0.811i)12-s + (2.45 + 1.41i)13-s + (2.81 − 3.07i)14-s + (1.67 + 0.432i)15-s + (2.36 − 4.10i)16-s + (2.76 − 4.79i)17-s + ⋯ |
L(s) = 1 | + (0.965 + 0.557i)2-s + (−0.968 − 0.249i)3-s + (0.121 + 0.210i)4-s − 0.447·5-s + (−0.795 − 0.781i)6-s + (0.215 − 0.976i)7-s − 0.843i·8-s + (0.875 + 0.483i)9-s + (−0.431 − 0.249i)10-s − 1.21i·11-s + (−0.0650 − 0.234i)12-s + (0.680 + 0.392i)13-s + (0.752 − 0.822i)14-s + (0.433 + 0.111i)15-s + (0.592 − 1.02i)16-s + (0.670 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21532 - 0.620687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21532 - 0.620687i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.67 + 0.432i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.569 + 2.58i)T \) |
good | 2 | \( 1 + (-1.36 - 0.788i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + 4.04iT - 11T^{2} \) |
| 13 | \( 1 + (-2.45 - 1.41i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.76 + 4.79i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (7.37 - 4.25i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.953iT - 23T^{2} \) |
| 29 | \( 1 + (-2.77 + 1.60i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.04 - 1.18i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.10 - 1.90i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.62 - 2.80i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.84 - 4.93i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.16 + 7.21i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.73 - 2.73i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.74 - 6.48i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.757 - 0.437i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.90 - 6.76i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.34iT - 71T^{2} \) |
| 73 | \( 1 + (-14.1 - 8.18i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.50 - 6.07i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.79 + 4.83i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.18 - 2.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.7 + 8.51i)T + (48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.60368551858452584130568506395, −10.81633646178853299102510317978, −9.972933240022811645171170532628, −8.359005631428340064316305809961, −7.27025022560095026668761874129, −6.42799720312052306799912778866, −5.62355091839086443498340118942, −4.49336571805372753370561482929, −3.69027905600655573797298691541, −0.877769943705403896899247443878,
2.13297141323560877277666612757, 3.79256179542093493558227121993, 4.64364658907551012558433064496, 5.55427522532643227021818100929, 6.55928099009092285298608808127, 8.030901077054702696562431192242, 9.022637890265299910101551189274, 10.47689566257714300426848970473, 11.04441319646955297238605553864, 12.02538263096549689974445704204