| L(s) = 1 | + (15.1 − 3.81i)3-s − 40.5i·5-s + 81.4i·7-s + (213. − 115. i)9-s − 499.·11-s − 262.·13-s + (−154. − 613. i)15-s + 565. i·17-s + 2.13e3i·19-s + (310. + 1.23e3i)21-s − 3.65e3·23-s + 1.47e3·25-s + (2.79e3 − 2.55e3i)27-s + 7.31e3i·29-s − 2.21e3i·31-s + ⋯ |
| L(s) = 1 | + (0.969 − 0.244i)3-s − 0.725i·5-s + 0.628i·7-s + (0.880 − 0.474i)9-s − 1.24·11-s − 0.430·13-s + (−0.177 − 0.703i)15-s + 0.474i·17-s + 1.35i·19-s + (0.153 + 0.609i)21-s − 1.43·23-s + 0.473·25-s + (0.737 − 0.675i)27-s + 1.61i·29-s − 0.414i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.356154051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356154051\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-15.1 + 3.81i)T \) |
| good | 5 | \( 1 + 40.5iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 81.4iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 499.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 262.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 565. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.13e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.65e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.31e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.21e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.00e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.86e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 8.28e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 3.92e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.49e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 6.36e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.09e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.87e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.48e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.71e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 7.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.84e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.48e5T + 8.58e9T^{2} \) |
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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
−10.49414124441827839278313107916, −9.811019127817385571789567937885, −8.693822923144566906459156595425, −8.218978723649414670695664579749, −7.31248159555116277460419578696, −5.91869760987054791174428063234, −4.92383409132516740409656442202, −3.66438278423693814944024523018, −2.47126847267134021184357864829, −1.46389045330974959423673306162,
0.26943015696211501423976511159, 2.19787945316370388259260786246, 2.95710142783520186780744872280, 4.13772504875395329481775228040, 5.19451469547382299380565187037, 6.78336118493382082982814041080, 7.49841765749351610621382937365, 8.289931902906358724458397275870, 9.447705339432828157875911025565, 10.28698395249524346919617629139
