| L(s) = 1 | + (−1.73 + i)2-s + (2.82 + 4.36i)3-s + (1.99 − 3.46i)4-s + (2.24 + 3.88i)5-s + (−9.25 − 4.73i)6-s + (−9.71 + 15.7i)7-s + 7.99i·8-s + (−11.0 + 24.6i)9-s + (−7.77 − 4.49i)10-s + (20.2 + 11.7i)11-s + (20.7 − 1.04i)12-s − 5.91i·13-s + (1.05 − 37.0i)14-s + (−10.6 + 20.7i)15-s + (−8 − 13.8i)16-s + (58.0 − 100. i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.543 + 0.839i)3-s + (0.249 − 0.433i)4-s + (0.200 + 0.347i)5-s + (−0.629 − 0.322i)6-s + (−0.524 + 0.851i)7-s + 0.353i·8-s + (−0.410 + 0.912i)9-s + (−0.245 − 0.142i)10-s + (0.555 + 0.320i)11-s + (0.499 − 0.0252i)12-s − 0.126i·13-s + (0.0201 − 0.706i)14-s + (−0.183 + 0.357i)15-s + (−0.125 − 0.216i)16-s + (0.828 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.738699 + 0.857692i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.738699 + 0.857692i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.73 - i)T \) |
| 3 | \( 1 + (-2.82 - 4.36i)T \) |
| 7 | \( 1 + (9.71 - 15.7i)T \) |
| good | 5 | \( 1 + (-2.24 - 3.88i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-20.2 - 11.7i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 5.91iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-58.0 + 100. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-8.02 + 4.63i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-107. + 62.1i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 207. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-122. - 70.8i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (149. + 259. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 508.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 391.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (40.2 + 69.7i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-258. - 149. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-102. + 177. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (543. - 313. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-51.3 + 89.0i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 46.9iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (228. + 131. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-533. - 924. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 270.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-443. - 768. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 219. iT - 9.12e5T^{2} \) |
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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
−15.88913300604632470656426320939, −14.87568367757269887191921869419, −13.99614214412162047344328218750, −12.15662791315647334146628145452, −10.61879230076114580300697668844, −9.525865785568021253436569562661, −8.694634070860015781760036032535, −6.96990952805599800968498455695, −5.22702618219366610770164725666, −2.88586236306103118510840540375,
1.23784660790389368779327579723, 3.49329648580107071537752523870, 6.39204332055216023316071157025, 7.71587853949401233833043539041, 8.934791279203136239396752783351, 10.14828171904260534420220637665, 11.71624433953989704752140757043, 12.91665516454284821754735993927, 13.73844187519680030961273535498, 15.15914935447740075822748510795
