| L(s) = 1 | + 2.63·5-s − 12.5i·7-s − 5.79i·11-s + 8.78·13-s + 30.1·17-s + 17.4i·19-s + 2.48i·23-s − 18.0·25-s + 26.4·29-s − 38.0i·31-s − 33.0i·35-s − 47.7·37-s − 53.3·41-s − 30.7i·43-s + 16.2i·47-s + ⋯ |
| L(s) = 1 | + 0.527·5-s − 1.79i·7-s − 0.527i·11-s + 0.675·13-s + 1.77·17-s + 0.916i·19-s + 0.108i·23-s − 0.722·25-s + 0.910·29-s − 1.22i·31-s − 0.945i·35-s − 1.29·37-s − 1.30·41-s − 0.714i·43-s + 0.345i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.123924609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.123924609\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.63T + 25T^{2} \) |
| 7 | \( 1 + 12.5iT - 49T^{2} \) |
| 11 | \( 1 + 5.79iT - 121T^{2} \) |
| 13 | \( 1 - 8.78T + 169T^{2} \) |
| 17 | \( 1 - 30.1T + 289T^{2} \) |
| 19 | \( 1 - 17.4iT - 361T^{2} \) |
| 23 | \( 1 - 2.48iT - 529T^{2} \) |
| 29 | \( 1 - 26.4T + 841T^{2} \) |
| 31 | \( 1 + 38.0iT - 961T^{2} \) |
| 37 | \( 1 + 47.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 53.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 30.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 107. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 62.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 19.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.13T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.83iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 159. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 39.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 60.5T + 9.40e3T^{2} \) |
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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
−9.775439849014747176058036740225, −8.374982247806938287482027339247, −7.81491134752391433140154719510, −6.91947234982246740336393479533, −6.04040490585524183681246875688, −5.20072351738704888286950526281, −3.87189455896222145922695881319, −3.43068744541868047413079277700, −1.65787183287388615058910534514, −0.67739469477376889910939236780,
1.43035711326549308185042646561, 2.50493107587629151424830985074, 3.40880543912367347158888244083, 4.95859996471841106504370806752, 5.54678226650505936944582368018, 6.29542758626059570201165894974, 7.29275013668584935409008275458, 8.517657102430528548806919919281, 8.815399977115984052556100837308, 9.856846003604512758128673006849
