L(s) = 1 | + 2-s + (1.73 − 0.0741i)3-s + 4-s + (−1.09 + 0.634i)5-s + (1.73 − 0.0741i)6-s + (0.151 + 2.64i)7-s + 8-s + (2.98 − 0.256i)9-s + (−1.09 + 0.634i)10-s + (2.57 + 4.46i)11-s + (1.73 − 0.0741i)12-s + (−2.55 − 2.53i)13-s + (0.151 + 2.64i)14-s + (−1.85 + 1.17i)15-s + 16-s − 5.00·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.999 − 0.0427i)3-s + 0.5·4-s + (−0.491 + 0.283i)5-s + (0.706 − 0.0302i)6-s + (0.0572 + 0.998i)7-s + 0.353·8-s + (0.996 − 0.0855i)9-s + (−0.347 + 0.200i)10-s + (0.776 + 1.34i)11-s + (0.499 − 0.0213i)12-s + (−0.709 − 0.704i)13-s + (0.0404 + 0.705i)14-s + (−0.478 + 0.304i)15-s + 0.250·16-s − 1.21·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.76554 + 0.713914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.76554 + 0.713914i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.73 + 0.0741i)T \) |
| 7 | \( 1 + (-0.151 - 2.64i)T \) |
| 13 | \( 1 + (2.55 + 2.53i)T \) |
good | 5 | \( 1 + (1.09 - 0.634i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.57 - 4.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 5.00T + 17T^{2} \) |
| 19 | \( 1 + (-3.30 + 5.71i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.70iT - 23T^{2} \) |
| 29 | \( 1 + (0.776 + 0.448i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.25 + 7.37i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.53iT - 37T^{2} \) |
| 41 | \( 1 + (-4.54 - 2.62i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.72 - 2.98i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.18 - 2.41i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (11.7 + 6.76i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 11.8iT - 59T^{2} \) |
| 61 | \( 1 + (-4.04 - 2.33i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.0 - 5.78i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.79 - 6.56i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.210 + 0.365i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.95 + 5.10i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.22iT - 83T^{2} \) |
| 89 | \( 1 + 8.30iT - 89T^{2} \) |
| 97 | \( 1 + (-6.21 - 10.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.13317793240622915188671948309, −9.699547056953578530290493552006, −9.268608173834954422409885537805, −8.043912984018314079576729370234, −7.26900393036792907586279452535, −6.46211446480808418498760918713, −4.95215052689185100857469923754, −4.20703751933381215831231667231, −2.89268113102455649582393751454, −2.13078198030104423971276757182,
1.47470250178415165700350607099, 3.13798166917281644138684295738, 3.96190105230076977285548376066, 4.67567792649286941141403624756, 6.25894798320124497581957064022, 7.16467333320631835489685272836, 8.004455268278923323562030644571, 8.846133305883490183857754184132, 9.844067970777488165998180716453, 10.80858122851078152727116083375