L(s) = 1 | + (−2.16 + 1.24i)2-s + (1.41 + 2.44i)3-s + (2.11 − 3.66i)4-s + i·5-s + (−6.10 − 3.52i)6-s + (−1.64 − 0.952i)7-s + 5.55i·8-s + (−2.49 + 4.32i)9-s + (−1.24 − 2.16i)10-s + (0.926 − 0.534i)11-s + 11.9·12-s + (1.40 − 3.32i)13-s + 4.75·14-s + (−2.44 + 1.41i)15-s + (−2.70 − 4.69i)16-s + (0.318 − 0.551i)17-s + ⋯ |
L(s) = 1 | + (−1.52 + 0.882i)2-s + (0.816 + 1.41i)3-s + (1.05 − 1.83i)4-s + 0.447i·5-s + (−2.49 − 1.43i)6-s + (−0.623 − 0.360i)7-s + 1.96i·8-s + (−0.831 + 1.44i)9-s + (−0.394 − 0.683i)10-s + (0.279 − 0.161i)11-s + 3.44·12-s + (0.388 − 0.921i)13-s + 1.27·14-s + (−0.632 + 0.364i)15-s + (−0.677 − 1.17i)16-s + (0.0772 − 0.133i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.253878 + 0.502645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.253878 + 0.502645i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 13 | \( 1 + (-1.40 + 3.32i)T \) |
good | 2 | \( 1 + (2.16 - 1.24i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.41 - 2.44i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (1.64 + 0.952i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.926 + 0.534i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.318 + 0.551i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.96 - 2.86i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.90 - 3.30i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.72 + 8.18i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.46iT - 31T^{2} \) |
| 37 | \( 1 + (-0.655 + 0.378i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.232 - 0.133i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.318 + 0.551i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.44iT - 47T^{2} \) |
| 53 | \( 1 + 6.99T + 53T^{2} \) |
| 59 | \( 1 + (0.641 + 0.370i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.09 - 3.63i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.01 - 4.04i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.45 - 4.88i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 3.71iT - 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + 5.11iT - 83T^{2} \) |
| 89 | \( 1 + (10.8 - 6.28i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.65 + 2.11i)T + (48.5 + 84.0i)T^{2} \) |
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
−15.53341881481221786280921006039, −14.83641727829813048398514160042, −13.67150442853710266738381457444, −11.15735950202928408425135072040, −10.03414202180356788996274922473, −9.656169692112918419661671502619, −8.467928918367636435124085718426, −7.39259802992141814342508989946, −5.74789291303059430250809435651, −3.44960781111466495319585531492,
1.47694879281105926563825759706, 2.99578752465532891473641690173, 6.70790455546909363989073532389, 7.73796348035818052618654332204, 8.926078177384048536639686941504, 9.387034785891927352362762674841, 11.20221260677500624781644267718, 12.28053742195412289460588610392, 12.94831004157138800030883729464, 14.22329102953425615021118316741