L(s) = 1 | + (2.5 + 4.33i)5-s + (11 − 19.0i)7-s + (4.5 − 7.79i)11-s + (−8.5 − 14.7i)13-s − 75·17-s − 4·19-s + (−91.5 − 158. i)23-s + (−12.5 + 21.6i)25-s + (−64.5 + 111. i)29-s + (93.5 + 161. i)31-s + 110·35-s − 34·37-s + (−132 − 228. i)41-s + (−221.5 + 383. i)43-s + (−304.5 + 527. i)47-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (0.593 − 1.02i)7-s + (0.123 − 0.213i)11-s + (−0.181 − 0.314i)13-s − 1.07·17-s − 0.0482·19-s + (−0.829 − 1.43i)23-s + (−0.100 + 0.173i)25-s + (−0.413 + 0.715i)29-s + (0.541 + 0.938i)31-s + 0.531·35-s − 0.151·37-s + (−0.502 − 0.870i)41-s + (−0.785 + 1.36i)43-s + (−0.945 + 1.63i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3700606791\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3700606791\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 7 | \( 1 + (-11 + 19.0i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-4.5 + 7.79i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (8.5 + 14.7i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 75T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4T + 6.85e3T^{2} \) |
| 23 | \( 1 + (91.5 + 158. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (64.5 - 111. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-93.5 - 161. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 34T + 5.06e4T^{2} \) |
| 41 | \( 1 + (132 + 228. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (221.5 - 383. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (304.5 - 527. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 228T + 1.48e5T^{2} \) |
| 59 | \( 1 + (30 + 51.9i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-227 + 393. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-122 - 211. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 444T + 3.57e5T^{2} \) |
| 73 | \( 1 - 398T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-174.5 + 302. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (519 - 898. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 852T + 7.04e5T^{2} \) |
| 97 | \( 1 + (457 - 791. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−9.385622168920437679731523821848, −8.397628126050123129068039757861, −7.83214622170171280007631056117, −6.78804189495378307992983748081, −6.38478529071942104113336604242, −5.07012624363549980551679499259, −4.40482675211002225245496698949, −3.41541412313086067282456607700, −2.30421208097459375534530624418, −1.19245964081166023620335598029,
0.07705602809717099220043430140, 1.73658514844953754449671948837, 2.26474847239114265376531246039, 3.68242256193953417426220752352, 4.66928225430683723737479724075, 5.41339706787110531867111322979, 6.17655481834346146560559062648, 7.12166496940863698586099927593, 8.141514862239300309393068210237, 8.655668778363272125702349603063