| L(s) = 1 | + (0.355 − 4.06i)2-s + (−0.359 − 5.18i)3-s + (−8.51 − 1.50i)4-s + (−6.07 + 9.38i)5-s + (−21.2 − 0.381i)6-s + (−6.05 + 8.64i)7-s + (−0.687 + 2.56i)8-s + (−26.7 + 3.72i)9-s + (36.0 + 28.0i)10-s + (−16.2 + 44.6i)11-s + (−4.72 + 44.7i)12-s + (−24.5 + 2.14i)13-s + (32.9 + 27.6i)14-s + (50.8 + 28.0i)15-s + (−54.8 − 19.9i)16-s + (−27.7 − 103. i)17-s + ⋯ |
| L(s) = 1 | + (0.125 − 1.43i)2-s + (−0.0692 − 0.997i)3-s + (−1.06 − 0.187i)4-s + (−0.543 + 0.839i)5-s + (−1.44 − 0.0259i)6-s + (−0.326 + 0.466i)7-s + (−0.0304 + 0.113i)8-s + (−0.990 + 0.138i)9-s + (1.13 + 0.886i)10-s + (−0.445 + 1.22i)11-s + (−0.113 + 1.07i)12-s + (−0.523 + 0.0458i)13-s + (0.629 + 0.528i)14-s + (0.875 + 0.483i)15-s + (−0.856 − 0.311i)16-s + (−0.395 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.158262 + 0.119408i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.158262 + 0.119408i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.359 + 5.18i)T \) |
| 5 | \( 1 + (6.07 - 9.38i)T \) |
| good | 2 | \( 1 + (-0.355 + 4.06i)T + (-7.87 - 1.38i)T^{2} \) |
| 7 | \( 1 + (6.05 - 8.64i)T + (-117. - 322. i)T^{2} \) |
| 11 | \( 1 + (16.2 - 44.6i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (24.5 - 2.14i)T + (2.16e3 - 381. i)T^{2} \) |
| 17 | \( 1 + (27.7 + 103. i)T + (-4.25e3 + 2.45e3i)T^{2} \) |
| 19 | \( 1 + (75.6 - 43.6i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-155. + 109. i)T + (4.16e3 - 1.14e4i)T^{2} \) |
| 29 | \( 1 + (23.0 - 19.3i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-32.5 + 184. i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (430. - 115. i)T + (4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-103. + 123. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (255. - 118. i)T + (5.11e4 - 6.09e4i)T^{2} \) |
| 47 | \( 1 + (-119. - 83.6i)T + (3.55e4 + 9.75e4i)T^{2} \) |
| 53 | \( 1 + (-173. - 173. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (206. - 74.9i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (66.4 + 376. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (27.1 + 310. i)T + (-2.96e5 + 5.22e4i)T^{2} \) |
| 71 | \( 1 + (560. + 323. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-455. - 122. i)T + (3.36e5 + 1.94e5i)T^{2} \) |
| 79 | \( 1 + (513. + 612. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (497. + 43.5i)T + (5.63e5 + 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-159. - 277. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-332. - 713. i)T + (-5.86e5 + 6.99e5i)T^{2} \) |
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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
−12.09399912660230317681214437100, −11.14279689350787256787560943080, −10.24376270250539857261931130411, −9.026489006373104666469297715089, −7.44284688230848746967750365951, −6.67908697085369189247087614576, −4.72565805050281104870518412415, −2.95095079736007881098604131533, −2.18556188411895180009125765056, −0.090563934242092518190192537497,
3.64483624273686393692467368660, 4.86131925731626306205342119895, 5.72695802853840680746700982726, 7.06820218859588467844632669485, 8.498147392799418897212075323631, 8.795445832086470016399857834579, 10.44848077486245554205704110599, 11.35600929915610778988980655600, 12.91040437728871509468750959916, 13.78037677039140004268488981640
