L(s) = 1 | + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−2.5 + 4.33i)5-s + (−7 − 12.1i)7-s − 7.99·8-s − 10·10-s + (−1.5 − 2.59i)11-s + (−23.5 + 40.7i)13-s + (14 − 24.2i)14-s + (−8 − 13.8i)16-s − 39·17-s + 32·19-s + (−10 − 17.3i)20-s + (3 − 5.19i)22-s + (49.5 − 85.7i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.377 − 0.654i)7-s − 0.353·8-s − 0.316·10-s + (−0.0411 − 0.0712i)11-s + (−0.501 + 0.868i)13-s + (0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s − 0.556·17-s + 0.386·19-s + (−0.111 − 0.193i)20-s + (0.0290 − 0.0503i)22-s + (0.448 − 0.777i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.456005631\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.456005631\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 7 | \( 1 + (7 + 12.1i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (23.5 - 40.7i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 39T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-49.5 + 85.7i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (25.5 + 44.1i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (41.5 - 71.8i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 314T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-54 + 93.5i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (149.5 + 258. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (265.5 + 459. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 564T + 1.48e5T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (115 + 199. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-134 + 232. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 120T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-369.5 - 639. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (543 + 940. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 120T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-821 - 1.42e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−9.753326134012964541545393152693, −8.886703673602922947762736687169, −7.926779977495317475216870007952, −6.93898204683007204984349142594, −6.65602663990705078774340325531, −5.36858874037411943107665746554, −4.36897766434579705916657929365, −3.55953139103198219434543786360, −2.30287016809113354940713719062, −0.40843114839121122645625447203,
0.986754177133864274022430160216, 2.42291469253189126017682938706, 3.29194227301197728120764433060, 4.48409181020971865935417330692, 5.35448569699092403208024011922, 6.15747063606377922517867111551, 7.40628084820660590996618090734, 8.300244597977675391919936349094, 9.345485164814553059047130600208, 9.764608830913919356627906031424