| L(s) = 1 | + (1.03 − 2.03i)2-s + (0.270 − 1.71i)3-s + (−0.718 − 0.989i)4-s + (−3.06 − 3.94i)5-s + (−3.20 − 2.32i)6-s + (0.410 + 0.410i)7-s + (6.26 − 0.992i)8-s + (−2.85 − 0.927i)9-s + (−11.2 + 2.15i)10-s + (0.00192 + 0.00591i)11-s + (−1.88 + 0.961i)12-s + (9.19 + 18.0i)13-s + (1.26 − 0.410i)14-s + (−7.58 + 4.18i)15-s + (5.99 − 18.4i)16-s + (−1.16 − 7.37i)17-s + ⋯ |
| L(s) = 1 | + (0.518 − 1.01i)2-s + (0.0903 − 0.570i)3-s + (−0.179 − 0.247i)4-s + (−0.613 − 0.789i)5-s + (−0.533 − 0.387i)6-s + (0.0586 + 0.0586i)7-s + (0.783 − 0.124i)8-s + (−0.317 − 0.103i)9-s + (−1.12 + 0.215i)10-s + (0.000174 + 0.000537i)11-s + (−0.157 + 0.0801i)12-s + (0.707 + 1.38i)13-s + (0.0901 − 0.0292i)14-s + (−0.505 + 0.278i)15-s + (0.374 − 1.15i)16-s + (−0.0687 − 0.433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.930705 - 1.32078i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.930705 - 1.32078i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.270 + 1.71i)T \) |
| 5 | \( 1 + (3.06 + 3.94i)T \) |
| good | 2 | \( 1 + (-1.03 + 2.03i)T + (-2.35 - 3.23i)T^{2} \) |
| 7 | \( 1 + (-0.410 - 0.410i)T + 49iT^{2} \) |
| 11 | \( 1 + (-0.00192 - 0.00591i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-9.19 - 18.0i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (1.16 + 7.37i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (3.79 - 5.21i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-35.1 - 17.8i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (5.20 + 7.16i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (22.1 + 16.0i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-2.08 + 1.06i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (14.5 - 44.8i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (39.9 - 39.9i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (91.4 + 14.4i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-12.8 + 81.2i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (48.4 + 15.7i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-8.83 - 27.1i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-4.58 - 28.9i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (38.0 - 27.6i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-94.5 - 48.1i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-2.24 - 3.09i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (26.0 - 4.11i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-26.2 + 8.53i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-185. - 29.3i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−13.38078673160469411683218695299, −12.94678264508324057741996800654, −11.53465309021795926365812446972, −11.44856139654401983630618936228, −9.488796789143274526387681252811, −8.229635828614447010235950334316, −6.92199677019748981909425089199, −4.89341158888074037556160412762, −3.54352851994312700990878068330, −1.58139787853591598938322299429,
3.39941473465543599056482510342, 4.93296584486699604716000675482, 6.26990264200112289218151379185, 7.44937405470030162875763415941, 8.576832337087067575988810028659, 10.51901024529652043712534777769, 10.96280558506873066784024831491, 12.75634822555072965340087777991, 13.93903188261504981255833701059, 15.05546749872370842575769325363
