| L(s) = 1 | + 1.41·5-s + 24i·13-s + 32.5i·17-s − 23·25-s + 1.41·29-s − 70i·37-s − 69.2i·41-s − 49·49-s − 103.·53-s − 22i·61-s + 33.9i·65-s + 96·73-s + 46i·85-s + 168. i·89-s − 144·97-s + ⋯ |
| L(s) = 1 | + 0.282·5-s + 1.84i·13-s + 1.91i·17-s − 0.920·25-s + 0.0487·29-s − 1.89i·37-s − 1.69i·41-s − 0.999·49-s − 1.94·53-s − 0.360i·61-s + 0.522i·65-s + 1.31·73-s + 0.541i·85-s + 1.89i·89-s − 1.48·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6972998128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6972998128\) |
| \(L(2)\) |
|
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 | \( 1 - 1.41T + 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 - 24iT - 169T^{2} \) |
| 17 | \( 1 - 32.5iT - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 1.41T + 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 + 70iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 69.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 103.T + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 + 22iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 168. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 144T + 9.40e3T^{2} \) |
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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
−9.254085497229185929248265822851, −8.468251183867451756198536950410, −7.69273795052874336591117886594, −6.72525936889767904456656062311, −6.18218578877772472100235833310, −5.31985803890213966103242703215, −4.17156508503479391566961883679, −3.73368506844486739178244947320, −2.18194333067412668526138470071, −1.60670615511753637354622513128,
0.16319160761093751898636623169, 1.31365068187300220348025650684, 2.76277005879972610911122811248, 3.22188906562215573384401901285, 4.64865654372599771548220061868, 5.23448687442913221682203019954, 6.08263385305934633375010398408, 6.90424808053979964882234791919, 7.898450500653818807549465135643, 8.202365239962739368080389969130
