L(s) = 1 | + (−0.347 − 1.96i)2-s + (−5.68 + 4.76i)3-s + (−3.75 + 1.36i)4-s + (18.4 + 6.70i)5-s + (11.3 + 9.53i)6-s + (−15.6 + 27.0i)7-s + (4 + 6.92i)8-s + (4.86 − 27.5i)9-s + (6.80 − 38.6i)10-s + (−5.32 − 9.23i)11-s + (14.8 − 25.6i)12-s + (−13.5 − 11.3i)13-s + (58.6 + 21.3i)14-s + (−136. + 49.7i)15-s + (12.2 − 10.2i)16-s + (4.26 + 24.1i)17-s + ⋯ |
L(s) = 1 | + (−0.122 − 0.696i)2-s + (−1.09 + 0.917i)3-s + (−0.469 + 0.171i)4-s + (1.64 + 0.599i)5-s + (0.773 + 0.648i)6-s + (−0.842 + 1.45i)7-s + (0.176 + 0.306i)8-s + (0.180 − 1.02i)9-s + (0.215 − 1.22i)10-s + (−0.146 − 0.253i)11-s + (0.356 − 0.618i)12-s + (−0.288 − 0.242i)13-s + (1.11 + 0.407i)14-s + (−2.35 + 0.855i)15-s + (0.191 − 0.160i)16-s + (0.0608 + 0.345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.766279 + 0.490585i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.766279 + 0.490585i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.347 + 1.96i)T \) |
| 19 | \( 1 + (-82.5 + 6.28i)T \) |
good | 3 | \( 1 + (5.68 - 4.76i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (-18.4 - 6.70i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (15.6 - 27.0i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (5.32 + 9.23i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (13.5 + 11.3i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-4.26 - 24.1i)T + (-4.61e3 + 1.68e3i)T^{2} \) |
| 23 | \( 1 + (-46.1 + 16.8i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (18.7 - 106. i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-147. + 256. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 68.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-39.8 + 33.4i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-273. - 99.5i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (47.7 - 270. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + (-318. + 116. i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (19.1 + 108. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (401. - 146. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (122. - 694. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (250. + 91.0i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-689. + 578. i)T + (6.75e4 - 3.83e5i)T^{2} \) |
| 79 | \( 1 + (-372. + 312. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (91.6 - 158. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-303. - 254. i)T + (1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-119. - 675. i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−16.20851321460227309782099357704, −14.95697278739465414800971869639, −13.43157206999043903606454849177, −12.21767696557828760238420708743, −10.94602727313602148853844689806, −9.910868783855234030949152142258, −9.278562274636402578645027490934, −6.11655185780760764477062036624, −5.36257678093405555117861929700, −2.72897573199322621498354974333,
0.989173392947323134565786618148, 5.15818443770317536821666744455, 6.37274377334070495189326026644, 7.24557993376303301568311904523, 9.457113143189875099453741136463, 10.42698348296670959848612238702, 12.36980414865654730697636948035, 13.42239973055720435589324412471, 13.90707449116893636640503317345, 16.19557293953252975290704247410