| L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 8.61i·5-s − 2.64·7-s − 2.82i·8-s + 12.1·10-s − 14.8i·11-s − 5.12·13-s − 3.74i·14-s + 4.00·16-s − 11.0i·17-s − 31.6·19-s + 17.2i·20-s + 20.9·22-s − 9.03i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.72i·5-s − 0.377·7-s − 0.353i·8-s + 1.21·10-s − 1.34i·11-s − 0.394·13-s − 0.267i·14-s + 0.250·16-s − 0.648i·17-s − 1.66·19-s + 0.861i·20-s + 0.954·22-s − 0.393i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3960763698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3960763698\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 + 8.61iT - 25T^{2} \) |
| 11 | \( 1 + 14.8iT - 121T^{2} \) |
| 13 | \( 1 + 5.12T + 169T^{2} \) |
| 17 | \( 1 + 11.0iT - 289T^{2} \) |
| 19 | \( 1 + 31.6T + 361T^{2} \) |
| 23 | \( 1 + 9.03iT - 529T^{2} \) |
| 29 | \( 1 - 38.7iT - 841T^{2} \) |
| 31 | \( 1 - 58.5T + 961T^{2} \) |
| 37 | \( 1 - 16.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 35.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 80.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 4.52iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 2.19iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 35.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 59.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 40.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 62.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 30.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 32.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 61.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 99.2T + 9.40e3T^{2} \) |
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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
−8.900157357093834615044974028510, −8.519168321521338399340058250532, −7.78130996039434130933203118145, −6.49146989130494277253930600955, −5.88582514447554127517232032749, −4.84564006011557930401146821003, −4.35022365130015522374493768436, −2.94328017708493379518549406653, −1.16416683118276305843081083323, −0.12767024013825028189195365526,
2.05222844895588263591106232372, 2.61897771789387897111790672426, 3.77553200160003547031239248290, 4.53308339968433656709209707081, 6.02578917451438003691875838371, 6.69212396703461290814576094469, 7.47405312474829221623551371687, 8.417823771537389038180623770958, 9.631810610609776820902142357804, 10.25968371362105175624448835823
