| L(s) = 1 | + (0.0302 − 0.403i)2-s + (2.45 − 1.67i)3-s + (1.81 + 0.273i)4-s + (−0.669 + 2.93i)5-s + (−0.600 − 1.04i)6-s + (−2.63 + 0.280i)7-s + (0.345 − 1.51i)8-s + (2.12 − 5.41i)9-s + (1.16 + 0.358i)10-s + (0.441 − 1.12i)11-s + (4.91 − 2.36i)12-s + (1.48 + 1.37i)13-s + (0.0337 + 1.06i)14-s + (3.26 + 8.31i)15-s + (2.91 + 0.897i)16-s + (−1.45 − 6.35i)17-s + ⋯ |
| L(s) = 1 | + (0.0213 − 0.285i)2-s + (1.41 − 0.965i)3-s + (0.907 + 0.136i)4-s + (−0.299 + 1.31i)5-s + (−0.245 − 0.424i)6-s + (−0.994 + 0.106i)7-s + (0.122 − 0.534i)8-s + (0.708 − 1.80i)9-s + (0.367 + 0.113i)10-s + (0.133 − 0.339i)11-s + (1.41 − 0.683i)12-s + (0.412 + 0.382i)13-s + (0.00901 + 0.285i)14-s + (0.842 + 2.14i)15-s + (0.727 + 0.224i)16-s + (−0.351 − 1.54i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.01937 - 0.664394i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01937 - 0.664394i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.63 - 0.280i)T \) |
| 43 | \( 1 + (6.40 - 1.39i)T \) |
| good | 2 | \( 1 + (-0.0302 + 0.403i)T + (-1.97 - 0.298i)T^{2} \) |
| 3 | \( 1 + (-2.45 + 1.67i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.669 - 2.93i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-0.441 + 1.12i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-1.48 - 1.37i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (1.45 + 6.35i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + (3.86 - 4.84i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (1.91 + 2.40i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (0.280 - 3.74i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (0.304 - 4.06i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 + 2.49T + 37T^{2} \) |
| 41 | \( 1 + (8.54 - 4.11i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-1.88 - 4.81i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (1.24 + 0.383i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-9.73 + 9.03i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (0.337 + 4.49i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (3.94 + 10.0i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (4.82 + 12.3i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (2.35 - 10.3i)T + (-65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + (-4.34 - 7.53i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.628 + 8.38i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-5.86 - 2.82i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (-9.84 - 12.3i)T + (-21.5 + 94.5i)T^{2} \) |
| 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
−11.78619086680677985808065737197, −10.73716244650980557211726666591, −9.795575838330262833518958168125, −8.646361531823500431645221568666, −7.64458101955076257274364904558, −6.65477554095522967841531918932, −6.56376673557429102449594447299, −3.50184300138050707320455217443, −3.06532313321810264892362327152, −2.00248209145969802258605554905,
2.11013309964353521107877909474, 3.53216753602876103756198229438, 4.42851122887827946884338930777, 5.83015039575149699755194733276, 7.15561001228774811741726670588, 8.427899396175872534445648277163, 8.709592065953959332523014820604, 9.901471874819836764086493357229, 10.54149413832689992060874870060, 11.89029008036822171433884492484
