| L(s) = 1 | + (−2.09 − 1.42i)2-s + (−1.40 − 1.30i)3-s + (1.62 + 4.13i)4-s + (−2.38 + 0.737i)5-s + (1.08 + 4.74i)6-s + (−1.40 − 2.24i)7-s + (1.38 − 6.04i)8-s + (0.0507 + 0.676i)9-s + (6.06 + 1.87i)10-s + (0.229 − 3.05i)11-s + (3.11 − 7.93i)12-s + (−0.101 + 0.0487i)13-s + (−0.270 + 6.71i)14-s + (4.32 + 2.08i)15-s + (−5.02 + 4.66i)16-s + (−0.565 + 0.0852i)17-s + ⋯ |
| L(s) = 1 | + (−1.48 − 1.01i)2-s + (−0.811 − 0.753i)3-s + (0.811 + 2.06i)4-s + (−1.06 + 0.329i)5-s + (0.442 + 1.93i)6-s + (−0.529 − 0.848i)7-s + (0.487 − 2.13i)8-s + (0.0169 + 0.225i)9-s + (1.91 + 0.591i)10-s + (0.0690 − 0.921i)11-s + (0.898 − 2.29i)12-s + (−0.0280 + 0.0135i)13-s + (−0.0724 + 1.79i)14-s + (1.11 + 0.537i)15-s + (−1.25 + 1.16i)16-s + (−0.137 + 0.0206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0144938 + 0.199815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0144938 + 0.199815i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.40 + 2.24i)T \) |
| good | 2 | \( 1 + (2.09 + 1.42i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (1.40 + 1.30i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (2.38 - 0.737i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.229 + 3.05i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (0.101 - 0.0487i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (0.565 - 0.0852i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (1.46 + 2.52i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.98 - 1.05i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-0.419 - 0.526i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-2.54 + 4.40i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.977 - 2.49i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.963 + 4.22i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (2.61 + 11.4i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-2.74 - 1.87i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (0.511 + 1.30i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (12.2 + 3.77i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (0.199 - 0.507i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (2.29 - 3.97i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.460 + 0.578i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-11.2 + 7.64i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-2.00 - 3.47i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.54 - 3.63i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.756 + 10.0i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−15.55591561089026364441445946591, −13.32841901821810912228240859879, −12.18402910830466401811197399405, −11.28966259415211654744462713153, −10.67827431734461019604101445704, −9.069480630541846488512270439970, −7.67253314574321525539962762796, −6.75957912289767243966140613883, −3.44061553199877036414226043807, −0.55519317140735132468507853254,
4.88833823434399393945210845094, 6.37561462120524571691675161644, 7.78989012217996655724482418015, 8.992198326967348316834123854126, 10.04537620578991465807733768927, 11.18153647871106623912106652069, 12.40663727078026221352259389115, 14.95184373522040682569813907456, 15.56135664232531945715865546992, 16.29621152566513525876648781505
