| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 6.37i·5-s − 2.64·7-s + 2.82i·8-s − 9.01·10-s − 16.8i·11-s + 17.5·13-s + 3.74i·14-s + 4.00·16-s − 13.4i·17-s + 34.1·19-s + 12.7i·20-s − 23.8·22-s − 3.55i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 1.27i·5-s − 0.377·7-s + 0.353i·8-s − 0.901·10-s − 1.53i·11-s + 1.34·13-s + 0.267i·14-s + 0.250·16-s − 0.790i·17-s + 1.79·19-s + 0.637i·20-s − 1.08·22-s − 0.154i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.693701810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693701810\) |
| \(L(2)\) |
|
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 | \( 1 + 6.37iT - 25T^{2} \) |
| 11 | \( 1 + 16.8iT - 121T^{2} \) |
| 13 | \( 1 - 17.5T + 169T^{2} \) |
| 17 | \( 1 + 13.4iT - 289T^{2} \) |
| 19 | \( 1 - 34.1T + 361T^{2} \) |
| 23 | \( 1 + 3.55iT - 529T^{2} \) |
| 29 | \( 1 + 33.9iT - 841T^{2} \) |
| 31 | \( 1 - 30.0T + 961T^{2} \) |
| 37 | \( 1 + 37.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 32.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 69.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 26.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 8.14iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 42.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 14.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 42.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 54.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 38.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 98.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 21.1T + 9.40e3T^{2} \) |
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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
−9.292254613516026488491105462605, −8.469676107354003268252357028272, −8.031789311116998476169283322384, −6.50750363066720924985850714618, −5.59431531545923794410302599340, −4.88833434807587513861968352638, −3.69508618135338570749710232344, −2.99586652909792156867853944499, −1.27165784504404257661504705276, −0.59524844036618488520007483849,
1.54660102184766907293316722711, 3.08304753255597622923305213947, 3.80725832176396601918174383234, 5.06173749767962241210132259478, 6.01576668631504430418370991345, 6.91424593072325885608831217499, 7.20184930596080108840334461222, 8.270941818828348633848210447306, 9.191545682458461440373152864513, 10.13504963235084078198040219453
