| L(s) = 1 | + (8.14 + 2.64i)3-s + (11.1 + 0.261i)5-s − 36.0i·7-s + (37.4 + 27.2i)9-s + (−35.6 + 25.8i)11-s + (1.58 − 2.17i)13-s + (90.3 + 31.7i)15-s + (−95.8 + 31.1i)17-s + (18.9 + 58.3i)19-s + (95.4 − 293. i)21-s + (23.4 + 32.2i)23-s + (124. + 5.84i)25-s + (97.1 + 133. i)27-s + (−46.7 + 143. i)29-s + (−6.06 − 18.6i)31-s + ⋯ |
| L(s) = 1 | + (1.56 + 0.509i)3-s + (0.999 + 0.0233i)5-s − 1.94i·7-s + (1.38 + 1.00i)9-s + (−0.976 + 0.709i)11-s + (0.0337 − 0.0464i)13-s + (1.55 + 0.545i)15-s + (−1.36 + 0.444i)17-s + (0.228 + 0.704i)19-s + (0.991 − 3.05i)21-s + (0.212 + 0.292i)23-s + (0.998 + 0.0467i)25-s + (0.692 + 0.952i)27-s + (−0.299 + 0.921i)29-s + (−0.0351 − 0.108i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0788i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.71049 + 0.106979i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.71049 + 0.106979i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-11.1 - 0.261i)T \) |
| good | 3 | \( 1 + (-8.14 - 2.64i)T + (21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 + 36.0iT - 343T^{2} \) |
| 11 | \( 1 + (35.6 - 25.8i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-1.58 + 2.17i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (95.8 - 31.1i)T + (3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-18.9 - 58.3i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-23.4 - 32.2i)T + (-3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (46.7 - 143. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (6.06 + 18.6i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (40.8 - 56.2i)T + (-1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (13.9 + 10.1i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 240. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (386. + 125. i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-451. - 146. i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-102. - 74.8i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (221. - 160. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-1.01e3 + 331. i)T + (2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-92.0 + 283. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (487. + 670. i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-263. + 810. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-803. + 260. i)T + (4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (105. - 76.8i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-1.03e3 - 334. i)T + (7.38e5 + 5.36e5i)T^{2} \) |
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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
−13.45683371285194148559062213701, −13.09307314178061290072963384459, −10.60081843871248409039393210205, −10.19989278371929632849424640455, −9.148835735885169342857433962973, −7.900620120362540835034835933528, −6.92205772785400092020032960380, −4.75096353309518165994550451429, −3.53035714989370308358834902719, −1.93426917070603962853032955355,
2.24240370943616459060922164349, 2.77007139866230237193416324023, 5.27019855773077358787026268217, 6.56806762473898012783683013763, 8.221464619496001065517151164697, 8.901381703225386503407652921568, 9.613636808486434176037035121221, 11.34508220600612050844041980161, 12.86877394083467294255256282811, 13.29363146116139216520566774007
