| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 0.242·7-s + 2.82i·8-s − 3i·11-s + 3.48·13-s + 0.343i·14-s + 4.00·16-s − 7.75i·17-s − 8.97·19-s − 4.24·22-s + 6.51i·23-s − 4.92i·26-s + 0.485·28-s − 7.02i·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.0346·7-s + 0.353i·8-s − 0.272i·11-s + 0.268·13-s + 0.0245i·14-s + 0.250·16-s − 0.456i·17-s − 0.472·19-s − 0.192·22-s + 0.283i·23-s − 0.189i·26-s + 0.0173·28-s − 0.242i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8399047419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8399047419\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.242T + 49T^{2} \) |
| 11 | \( 1 + 3iT - 121T^{2} \) |
| 13 | \( 1 - 3.48T + 169T^{2} \) |
| 17 | \( 1 + 7.75iT - 289T^{2} \) |
| 19 | \( 1 + 8.97T + 361T^{2} \) |
| 23 | \( 1 - 6.51iT - 529T^{2} \) |
| 29 | \( 1 + 7.02iT - 841T^{2} \) |
| 31 | \( 1 - 29.2T + 961T^{2} \) |
| 37 | \( 1 + 36.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 57.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 11.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 6.51iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 26.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 84.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 57.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 97.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 72.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 106.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 78.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 25.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 31.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 110.T + 9.40e3T^{2} \) |
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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
−9.049642157260698752808475946515, −8.426883997224659599865676931551, −7.48512146560189350868313718924, −6.49125876772296984098575687366, −5.55283502754162324285764946326, −4.62019532134661869861351259134, −3.67325771943670317703765043761, −2.73834150845161479072475988387, −1.59030917801155097535221611925, −0.24692414202564874116458044714,
1.37229603333847437979186981095, 2.83149497860749755987219135550, 4.02324751665164782260258772679, 4.80383897796585852962680861201, 5.82342484592016201394506442862, 6.54271052446793336158447962890, 7.31806937624959456637646633977, 8.272467857349114383567552585221, 8.762002139551142010642157199985, 9.794089498044854922979178438865
