| L(s) = 1 | + 2-s + 4.47i·3-s − 3·4-s + 4.47i·6-s − 7·7-s − 7·8-s − 11.0·9-s + 2·11-s − 13.4i·12-s − 13.4i·13-s − 7·14-s + 5·16-s + 26.8i·17-s − 11.0·18-s + 13.4i·19-s + ⋯ |
| L(s) = 1 | + 0.5·2-s + 1.49i·3-s − 0.750·4-s + 0.745i·6-s − 7-s − 0.875·8-s − 1.22·9-s + 0.181·11-s − 1.11i·12-s − 1.03i·13-s − 0.5·14-s + 0.312·16-s + 1.57i·17-s − 0.611·18-s + 0.706i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.783303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.783303i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - T + 4T^{2} \) |
| 3 | \( 1 - 4.47iT - 9T^{2} \) |
| 11 | \( 1 - 2T + 121T^{2} \) |
| 13 | \( 1 + 13.4iT - 169T^{2} \) |
| 17 | \( 1 - 26.8iT - 289T^{2} \) |
| 19 | \( 1 - 13.4iT - 361T^{2} \) |
| 23 | \( 1 + 26T + 529T^{2} \) |
| 29 | \( 1 + 22T + 841T^{2} \) |
| 31 | \( 1 - 53.6iT - 961T^{2} \) |
| 37 | \( 1 + 14T + 1.36e3T^{2} \) |
| 41 | \( 1 - 26.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34T + 1.84e3T^{2} \) |
| 47 | \( 1 + 26.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 34T + 2.80e3T^{2} \) |
| 59 | \( 1 - 40.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 93.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 14T + 4.48e3T^{2} \) |
| 71 | \( 1 - 62T + 5.04e3T^{2} \) |
| 73 | \( 1 + 53.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 38T + 6.24e3T^{2} \) |
| 83 | \( 1 - 40.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 26.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 26.8iT - 9.40e3T^{2} \) |
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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
−12.91536394992629023881575693461, −12.25423672387255236361291503470, −10.57461699136790237582817836570, −10.08304732460621985622511326694, −9.165939351648076865556463342584, −8.208358749615457412753892010061, −6.15150899112071731328165737041, −5.29010600518441835995867457971, −3.95647343698359600242866392631, −3.40473490249149856042492731357,
0.40019394589696948159491333001, 2.50371016408415745461142317959, 4.09769343200612127736577170691, 5.70072161390158234500649530094, 6.67164245295287379208520995705, 7.57536351186365520859072305526, 8.985239900909979527651130182768, 9.670426737687766323550471917527, 11.56866256009431629392314537405, 12.20647871316742965032368694329
