| L(s) = 1 | + (−0.333 − 1.97i)2-s + (−0.927 + 3.46i)3-s + (−3.77 + 1.31i)4-s + (−1.39 − 4.80i)5-s + (7.13 + 0.675i)6-s + (6.90 − 1.16i)7-s + (3.85 + 7.01i)8-s + (−3.32 − 1.91i)9-s + (−9.00 + 4.34i)10-s + (7.98 − 4.61i)11-s + (−1.04 − 14.2i)12-s + (12.7 − 12.7i)13-s + (−4.59 − 13.2i)14-s + (17.9 − 0.368i)15-s + (12.5 − 9.92i)16-s + (−0.0413 + 0.154i)17-s + ⋯ |
| L(s) = 1 | + (−0.166 − 0.986i)2-s + (−0.309 + 1.15i)3-s + (−0.944 + 0.328i)4-s + (−0.278 − 0.960i)5-s + (1.18 + 0.112i)6-s + (0.986 − 0.166i)7-s + (0.481 + 0.876i)8-s + (−0.369 − 0.213i)9-s + (−0.900 + 0.434i)10-s + (0.726 − 0.419i)11-s + (−0.0870 − 1.19i)12-s + (0.977 − 0.977i)13-s + (−0.327 − 0.944i)14-s + (1.19 − 0.0245i)15-s + (0.784 − 0.620i)16-s + (−0.00243 + 0.00907i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.07489 - 0.548452i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07489 - 0.548452i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.333 + 1.97i)T \) |
| 5 | \( 1 + (1.39 + 4.80i)T \) |
| 7 | \( 1 + (-6.90 + 1.16i)T \) |
| good | 3 | \( 1 + (0.927 - 3.46i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (-7.98 + 4.61i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-12.7 + 12.7i)T - 169iT^{2} \) |
| 17 | \( 1 + (0.0413 - 0.154i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-16.3 - 9.43i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (7.72 + 28.8i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 12.2iT - 841T^{2} \) |
| 31 | \( 1 + (-23.7 - 41.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (8.80 + 32.8i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 5.88iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (32.4 - 32.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-10.3 - 38.5i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-17.5 + 65.3i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (12.6 - 7.29i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-58.5 - 33.7i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-18.5 + 69.0i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 38.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (33.0 + 8.85i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (51.1 - 88.5i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (57.8 + 57.8i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (74.9 - 129. i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-5.13 - 5.13i)T + 9.40e3iT^{2} \) |
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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
−12.46990072756006650107338614755, −11.53770704255484406912099464862, −10.77995960980593630845977317288, −9.886447527129575829490742505386, −8.739133118483601557903268966915, −8.076969771054808284526772185820, −5.49715385844835212807350367658, −4.56178975332587723430998435526, −3.63473919978385073813818232952, −1.13915344337405160222151737681,
1.50049213805196984708309890219, 4.09022441851310420535128424446, 5.78200506809368211200321533991, 6.82871862862703069819250114813, 7.40305578056688615012746947424, 8.485656378946335139477305550738, 9.792340832197684702047215818425, 11.37469200933247933870070913427, 11.90176329171130430257480942427, 13.52156177594246854391309486152
