| L(s) = 1 | + (0.628 + 3.56i)2-s + (−2.29 + 1.92i)3-s + (−4.78 + 1.74i)4-s + (−14.2 − 5.18i)5-s + (−8.31 − 6.97i)6-s + (−12.3 + 21.3i)7-s + (5.26 + 9.12i)8-s + (1.56 − 8.86i)9-s + (9.51 − 53.9i)10-s + (−22.6 − 39.2i)11-s + (7.63 − 13.2i)12-s + (64.8 + 54.4i)13-s + (−83.8 − 30.5i)14-s + (42.7 − 15.5i)15-s + (−60.3 + 50.6i)16-s + (4.59 + 26.0i)17-s + ⋯ |
| L(s) = 1 | + (0.222 + 1.25i)2-s + (−0.442 + 0.371i)3-s + (−0.597 + 0.217i)4-s + (−1.27 − 0.463i)5-s + (−0.565 − 0.474i)6-s + (−0.665 + 1.15i)7-s + (0.232 + 0.403i)8-s + (0.0578 − 0.328i)9-s + (0.300 − 1.70i)10-s + (−0.621 − 1.07i)11-s + (0.183 − 0.318i)12-s + (1.38 + 1.16i)13-s + (−1.60 − 0.582i)14-s + (0.735 − 0.267i)15-s + (−0.943 + 0.791i)16-s + (0.0655 + 0.371i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.112119 - 0.815683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112119 - 0.815683i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.29 - 1.92i)T \) |
| 19 | \( 1 + (21.7 - 79.9i)T \) |
| good | 2 | \( 1 + (-0.628 - 3.56i)T + (-7.51 + 2.73i)T^{2} \) |
| 5 | \( 1 + (14.2 + 5.18i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (12.3 - 21.3i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (22.6 + 39.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-64.8 - 54.4i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-4.59 - 26.0i)T + (-4.61e3 + 1.68e3i)T^{2} \) |
| 23 | \( 1 + (-14.5 + 5.30i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (43.1 - 244. i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-118. + 205. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 103.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-137. + 115. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (259. + 94.5i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-7.48 + 42.4i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + (-12.5 + 4.55i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (10.2 + 58.2i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-637. + 231. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (127. - 725. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (303. + 110. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-275. + 231. i)T + (6.75e4 - 3.83e5i)T^{2} \) |
| 79 | \( 1 + (182. - 152. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (229. - 397. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-819. - 688. i)T + (1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-92.1 - 522. i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−15.75039404191089455749770605629, −14.67305896760273434938835010529, −13.20863972894281346733469821497, −11.89879352892137952811700782624, −10.98859106077974905013184727237, −8.869723520963369139648107006976, −8.156396298347433446916391276293, −6.44793728210403263866659956085, −5.52586493037715904154335338246, −3.88767283962073454710991509950,
0.58779483937009052706199184009, 3.12085766981622827971979219354, 4.35165047023306349078334803680, 6.82518695173834477993682542348, 7.79886570856927027164901506487, 10.08519463200077578501734610661, 10.85050418660849728906293347117, 11.67758157469931570264027986570, 12.88564540881861195895637041034, 13.43425565526932757847706704929
