| L(s) = 1 | + 2.75i·2-s + 8.40·4-s + 21.9i·5-s + 63.3i·7-s + 67.2i·8-s − 60.5·10-s + 1.17·11-s − 173.·13-s − 174.·14-s − 51.0·16-s − 469.·17-s + 27.8i·19-s + 184. i·20-s + 3.22i·22-s − 79.3·23-s + ⋯ |
| L(s) = 1 | + 0.689i·2-s + 0.525·4-s + 0.879i·5-s + 1.29i·7-s + 1.05i·8-s − 0.605·10-s + 0.00967·11-s − 1.02·13-s − 0.890·14-s − 0.199·16-s − 1.62·17-s + 0.0770i·19-s + 0.461i·20-s + 0.00666i·22-s − 0.150·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.390680586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.390680586\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (1.43e3 - 1.16e3i)T \) |
| good | 2 | \( 1 - 2.75iT - 16T^{2} \) |
| 5 | \( 1 - 21.9iT - 625T^{2} \) |
| 7 | \( 1 - 63.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 1.17T + 1.46e4T^{2} \) |
| 13 | \( 1 + 173.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 469.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 27.8iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 79.3T + 2.79e5T^{2} \) |
| 29 | \( 1 + 696. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.19e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.53e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.45e3T + 2.82e6T^{2} \) |
| 47 | \( 1 + 1.69e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.99e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.56e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 5.44e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 4.93e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 9.66e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.60e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 1.14e4T + 3.89e7T^{2} \) |
| 83 | \( 1 + 5.52e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 2.55e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.65e3T + 8.85e7T^{2} \) |
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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
−11.35553586669231688932491229101, −10.42495817781466001925199218186, −9.280635211286271778092591656668, −8.346906164782926294869574750543, −7.34683450225663686233472956699, −6.51929142824768957088495585550, −5.81686684532088895166260353279, −4.63198691943063952311115564165, −2.72777023059354519290006054354, −2.25136198510256681262273227916,
0.36146952907815603256000221237, 1.44132666978241931236278331825, 2.73339645624159899700040067304, 4.11617776886320146111757160362, 4.87290269486201538286050191051, 6.55200739198549285486506178318, 7.20208219607702051152885269239, 8.346648020718751291400193294110, 9.489622851086853353771490570235, 10.27869924366900502979417290884
