L(s) = 1 | + (−1.52 − 1.81i)2-s + (−0.439 − 2.49i)3-s + (−0.623 + 3.53i)4-s + (−1.04 + 0.184i)5-s + (−3.84 + 4.58i)6-s + (−0.591 − 2.57i)7-s + (3.26 − 1.88i)8-s + (−3.18 + 1.16i)9-s + (1.92 + 1.61i)10-s − 2.17·11-s + 9.08·12-s + (4.74 + 3.98i)13-s + (−3.77 + 4.99i)14-s + (0.917 + 2.52i)15-s + (−1.61 − 0.588i)16-s + (1.08 − 2.97i)17-s + ⋯ |
L(s) = 1 | + (−1.07 − 1.28i)2-s + (−0.253 − 1.43i)3-s + (−0.311 + 1.76i)4-s + (−0.467 + 0.0824i)5-s + (−1.56 + 1.87i)6-s + (−0.223 − 0.974i)7-s + (1.15 − 0.665i)8-s + (−1.06 + 0.386i)9-s + (0.607 + 0.510i)10-s − 0.656·11-s + 2.62·12-s + (1.31 + 1.10i)13-s + (−1.00 + 1.33i)14-s + (0.236 + 0.651i)15-s + (−0.404 − 0.147i)16-s + (0.262 − 0.720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.178087 + 0.362561i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.178087 + 0.362561i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.591 + 2.57i)T \) |
| 19 | \( 1 + (2.57 + 3.51i)T \) |
good | 2 | \( 1 + (1.52 + 1.81i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (0.439 + 2.49i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (1.04 - 0.184i)T + (4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + 2.17T + 11T^{2} \) |
| 13 | \( 1 + (-4.74 - 3.98i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.08 + 2.97i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (4.58 + 3.84i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.85 + 0.327i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.545 - 0.945i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.26 + 3.61i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.49 + 3.77i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (4.46 + 1.62i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.63 - 4.48i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (1.86 + 0.328i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-9.40 - 3.42i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.67 + 7.95i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-9.19 + 10.9i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.22 + 8.87i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-3.68 + 0.648i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.99 - 5.49i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-8.54 - 4.93i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.14 - 6.49i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.13 + 12.0i)T + (-91.1 + 33.1i)T^{2} \) |
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
−12.39753482781848467973095521780, −11.39044764605321757590170824751, −10.90051923011729591487613246808, −9.579499000262112365531195387059, −8.312821888951373586056730318014, −7.51773178210447787674590425405, −6.43795045213455444485276896242, −3.88729773565505927305122091253, −2.15381244638495666907614637675, −0.61152147060681640547688238642,
3.75037833464762693024362985515, 5.51339711492188987594581211275, 6.03072503249940143700816017122, 8.027175690369944214823895031200, 8.466862230928425251872017583149, 9.744115546772427045789734490221, 10.30187101214909844389931742984, 11.46357037280792581933053584087, 12.98495806458507061541735826527, 14.74574867841517582383671159273