| L(s) = 1 | + (0.224 − 0.0949i)2-s + (0.463 + 1.66i)3-s + (−1.35 + 1.39i)4-s + (−2.45 + 1.73i)5-s + (0.262 + 0.330i)6-s + (−1.87 + 0.726i)7-s + (−0.346 + 0.895i)8-s + (−2.56 + 1.54i)9-s + (−0.385 + 0.622i)10-s + (−0.633 + 0.838i)11-s + (−2.95 − 1.60i)12-s + (−0.223 + 1.44i)13-s + (−0.351 + 0.341i)14-s + (−4.03 − 3.28i)15-s + (−0.112 − 3.64i)16-s + (7.33 + 2.58i)17-s + ⋯ |
| L(s) = 1 | + (0.158 − 0.0671i)2-s + (0.267 + 0.963i)3-s + (−0.675 + 0.696i)4-s + (−1.09 + 0.776i)5-s + (0.107 + 0.134i)6-s + (−0.708 + 0.274i)7-s + (−0.122 + 0.316i)8-s + (−0.856 + 0.516i)9-s + (−0.121 + 0.196i)10-s + (−0.190 + 0.252i)11-s + (−0.852 − 0.464i)12-s + (−0.0620 + 0.399i)13-s + (−0.0940 + 0.0911i)14-s + (−1.04 − 0.848i)15-s + (−0.0280 − 0.911i)16-s + (1.77 + 0.626i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.279410 - 0.469229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.279410 - 0.469229i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.463 - 1.66i)T \) |
| 103 | \( 1 + (-9.27 + 4.11i)T \) |
| good | 2 | \( 1 + (-0.224 + 0.0949i)T + (1.39 - 1.43i)T^{2} \) |
| 5 | \( 1 + (2.45 - 1.73i)T + (1.66 - 4.71i)T^{2} \) |
| 7 | \( 1 + (1.87 - 0.726i)T + (5.17 - 4.71i)T^{2} \) |
| 11 | \( 1 + (0.633 - 0.838i)T + (-3.01 - 10.5i)T^{2} \) |
| 13 | \( 1 + (0.223 - 1.44i)T + (-12.3 - 3.94i)T^{2} \) |
| 17 | \( 1 + (-7.33 - 2.58i)T + (13.2 + 10.6i)T^{2} \) |
| 19 | \( 1 + (-2.06 + 0.127i)T + (18.8 - 2.33i)T^{2} \) |
| 23 | \( 1 + (1.08 + 8.77i)T + (-22.3 + 5.61i)T^{2} \) |
| 29 | \( 1 + (2.37 - 5.15i)T + (-18.8 - 22.0i)T^{2} \) |
| 31 | \( 1 + (-2.18 + 0.0671i)T + (30.9 - 1.90i)T^{2} \) |
| 37 | \( 1 + (-6.41 - 7.03i)T + (-3.41 + 36.8i)T^{2} \) |
| 41 | \( 1 + (11.9 + 1.11i)T + (40.3 + 7.53i)T^{2} \) |
| 43 | \( 1 + (-0.824 - 2.59i)T + (-35.0 + 24.8i)T^{2} \) |
| 47 | \( 1 - 0.502T + 47T^{2} \) |
| 53 | \( 1 + (-2.84 - 4.29i)T + (-20.6 + 48.8i)T^{2} \) |
| 59 | \( 1 + (-1.24 + 3.21i)T + (-43.6 - 39.7i)T^{2} \) |
| 61 | \( 1 + (2.81 + 3.28i)T + (-9.35 + 60.2i)T^{2} \) |
| 67 | \( 1 + (-6.74 + 1.04i)T + (63.8 - 20.3i)T^{2} \) |
| 71 | \( 1 + (12.4 + 8.83i)T + (23.5 + 66.9i)T^{2} \) |
| 73 | \( 1 + (9.97 + 0.924i)T + (71.7 + 13.4i)T^{2} \) |
| 79 | \( 1 + (-4.59 + 3.25i)T + (26.2 - 74.5i)T^{2} \) |
| 83 | \( 1 + (7.69 + 1.19i)T + (79.0 + 25.1i)T^{2} \) |
| 89 | \( 1 + (0.689 - 2.42i)T + (-75.6 - 46.8i)T^{2} \) |
| 97 | \( 1 + (4.85 - 5.66i)T + (-14.8 - 95.8i)T^{2} \) |
| 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
−10.42024545449947552584831226960, −9.902515659027637023419884965218, −8.928243569426367322942287466677, −8.154855002671051190494804118908, −7.53920473312012972823542580348, −6.34456063395463561152760035382, −5.08593182504187242026579061013, −4.21893624857382498870940375626, −3.34734473441459975516583138340, −2.92529166267359493984433722308,
0.27440623600593188099621908946, 1.22304864596829994337793949094, 3.20423147799608816783677177075, 3.94405957202196687865843111403, 5.34427960086724618993063675499, 5.81695232768950070737140874145, 7.18576000678113508693532824430, 7.79718616587834394188760192132, 8.525162465164963536074775450918, 9.525333198747786154791752539563
