| L(s) = 1 | + (−1.61 + 1.17i)2-s + (0.927 − 2.85i)3-s + (1.23 − 3.80i)4-s + (11.1 − 0.115i)5-s + (1.85 + 5.70i)6-s + 5.62·7-s + (2.47 + 7.60i)8-s + (−7.28 − 5.29i)9-s + (−17.9 + 13.3i)10-s + (11.3 − 8.26i)11-s + (−9.70 − 7.05i)12-s + (3.46 + 2.51i)13-s + (−9.09 + 6.60i)14-s + (10.0 − 32.0i)15-s + (−12.9 − 9.40i)16-s + (−7.53 − 23.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.999 − 0.0103i)5-s + (0.126 + 0.388i)6-s + 0.303·7-s + (0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.567 + 0.421i)10-s + (0.311 − 0.226i)11-s + (−0.233 − 0.169i)12-s + (0.0739 + 0.0537i)13-s + (−0.173 + 0.126i)14-s + (0.172 − 0.550i)15-s + (−0.202 − 0.146i)16-s + (−0.107 − 0.330i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.58829 - 0.390328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58829 - 0.390328i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.61 - 1.17i)T \) |
| 3 | \( 1 + (-0.927 + 2.85i)T \) |
| 5 | \( 1 + (-11.1 + 0.115i)T \) |
| good | 7 | \( 1 - 5.62T + 343T^{2} \) |
| 11 | \( 1 + (-11.3 + 8.26i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-3.46 - 2.51i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (7.53 + 23.1i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (6.90 + 21.2i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-165. + 120. i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-52.2 + 160. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-64.7 - 199. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-75.9 - 55.1i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (110. + 80.4i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 232.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-66.6 + 205. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-70.1 + 215. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (364. + 265. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (632. - 459. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-61.3 - 188. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (168. - 517. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (290. - 211. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (10.6 - 32.7i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-374. - 1.15e3i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (1.04e3 - 756. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-8.92 + 27.4i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−12.60612920657499532240276533509, −11.31868357518907550463133643707, −10.29637248576431924501969054981, −9.161425640482016907199511558281, −8.427404462015466432060448074111, −7.04843768705893467045417097128, −6.24176882276809779820210421870, −4.94611025497409222342797651060, −2.60387370843089757922783137795, −1.09528497788674332712177344649,
1.53913261105045436801308615954, 3.07066793471123357662495610534, 4.71173628278720835341124622262, 6.09227431078763663477074324208, 7.52486330986454002503904472614, 8.883903592318987226998939876127, 9.504888766192386663098427295132, 10.51370793784126823928546080976, 11.29711833437260280579899418177, 12.60759437924337732286767239117
