L(s) = 1 | + (1.48 − 0.859i)2-s + (−4.11 − 3.16i)3-s + (−2.52 + 4.36i)4-s + (−2.5 − 4.33i)5-s + (−8.85 − 1.17i)6-s + (2.28 + 18.3i)7-s + 22.4i·8-s + (6.94 + 26.0i)9-s + (−7.44 − 4.29i)10-s + (49.7 + 28.7i)11-s + (24.2 − 10.0i)12-s − 44.6i·13-s + (19.2 + 25.4i)14-s + (−3.41 + 25.7i)15-s + (−0.881 − 1.52i)16-s + (−54.8 + 95.0i)17-s + ⋯ |
L(s) = 1 | + (0.526 − 0.304i)2-s + (−0.792 − 0.609i)3-s + (−0.315 + 0.545i)4-s + (−0.223 − 0.387i)5-s + (−0.602 − 0.0798i)6-s + (0.123 + 0.992i)7-s + 0.991i·8-s + (0.257 + 0.966i)9-s + (−0.235 − 0.135i)10-s + (1.36 + 0.787i)11-s + (0.582 − 0.240i)12-s − 0.952i·13-s + (0.366 + 0.485i)14-s + (−0.0587 + 0.443i)15-s + (−0.0137 − 0.0238i)16-s + (−0.783 + 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.963166 + 0.634994i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.963166 + 0.634994i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.11 + 3.16i)T \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
| 7 | \( 1 + (-2.28 - 18.3i)T \) |
good | 2 | \( 1 + (-1.48 + 0.859i)T + (4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-49.7 - 28.7i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 44.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (54.8 - 95.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (79.5 - 45.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (27.1 - 15.6i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 183. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-123. - 71.2i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (114. + 199. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 185.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 22.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (116. + 201. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-400. - 230. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-229. + 397. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (248. - 143. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-416. + 720. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 471. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-469. - 270. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (175. + 304. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 866.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-519. - 899. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 323. iT - 9.12e5T^{2} \) |
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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
−12.90263895766722843548369138819, −12.49877502334576322216986055070, −11.83801083039424095823518185349, −10.68228265009911950201883875709, −8.907953449950020657323871281399, −8.040761934317482655397429381533, −6.48372788770320770714185867796, −5.25766258936561345273329370372, −4.02416101198493797787879080531, −1.94177179033482876521612735826,
0.62556341331199104973002430029, 3.97166299699308696989679954574, 4.61542095342670073024749159459, 6.30910737658951056858708721878, 6.83887523346271990029981170964, 9.027342341953038973084671073102, 9.964424737790061477639734421470, 11.11429152847817663241571834485, 11.74660805086640092678625560274, 13.45456667631054290167708178646