| L(s) = 1 | + (−0.870 + 1.19i)2-s + (0.435 + 1.67i)3-s + (−0.0590 − 0.181i)4-s + (1.02 − 0.590i)5-s + (−2.38 − 0.936i)6-s + (0.114 + 0.0243i)7-s + (−2.54 − 0.827i)8-s + (−2.62 + 1.46i)9-s + (−0.182 + 1.73i)10-s + (1.65 − 1.83i)11-s + (0.278 − 0.178i)12-s + (−0.591 + 1.32i)13-s + (−0.128 + 0.115i)14-s + (1.43 + 1.45i)15-s + (3.51 − 2.55i)16-s + (0.129 + 0.144i)17-s + ⋯ |
| L(s) = 1 | + (−0.615 + 0.846i)2-s + (0.251 + 0.967i)3-s + (−0.0295 − 0.0908i)4-s + (0.457 − 0.264i)5-s + (−0.974 − 0.382i)6-s + (0.0432 + 0.00920i)7-s + (−0.900 − 0.292i)8-s + (−0.873 + 0.486i)9-s + (−0.0577 + 0.549i)10-s + (0.498 − 0.553i)11-s + (0.0804 − 0.0514i)12-s + (−0.163 + 0.368i)13-s + (−0.0344 + 0.0310i)14-s + (0.370 + 0.376i)15-s + (0.878 − 0.638i)16-s + (0.0314 + 0.0349i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.438918 + 0.709879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.438918 + 0.709879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.435 - 1.67i)T \) |
| 31 | \( 1 + (1.70 + 5.30i)T \) |
| good | 2 | \( 1 + (0.870 - 1.19i)T + (-0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (-1.02 + 0.590i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.114 - 0.0243i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-1.65 + 1.83i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.591 - 1.32i)T + (-8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-0.129 - 0.144i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.86 + 2.61i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.06 + 3.27i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-7.42 - 5.39i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (7.34 + 4.24i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.19 + 0.336i)T + (40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-2.97 - 6.68i)T + (-28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (-2.01 - 2.77i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (6.25 - 1.33i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (7.45 - 0.783i)T + (57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + 11.2iT - 61T^{2} \) |
| 67 | \( 1 + (3.96 + 6.86i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.31 + 15.6i)T + (-64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-6.58 - 5.93i)T + (7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-1.39 + 1.25i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.267 + 2.54i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-3.58 - 11.0i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.44 - 7.53i)T + (-78.4 + 57.0i)T^{2} \) |
| 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
−14.59207302230560709074520021054, −13.73104429288742807094958612712, −12.16509887346453528689536709276, −10.97120921166238323380764351354, −9.509687799339673632898127530677, −9.039657188126363150463053291138, −7.86231561000749003312506501263, −6.41799017500555663208826844123, −5.08474803853242493653929082028, −3.26959401997153799988660966700,
1.53171711768334973908864962503, 3.00488428551712406588706185127, 5.67769788866834810426443808152, 6.92822932945931218973559729160, 8.290417709057241757844768560761, 9.496974649882228206471880185322, 10.34649892723231972360951703725, 11.76692904814724998714597196873, 12.24971782169269271319040828679, 13.70402595717471099954624300102
