| L(s) = 1 | + (−4.88 + 8.45i)5-s + (−68.3 − 118. i)7-s + (326. + 565. i)11-s + (−125. + 216. i)13-s − 249.·17-s − 1.75e3·19-s + (−827. + 1.43e3i)23-s + (1.51e3 + 2.62e3i)25-s + (2.12e3 + 3.67e3i)29-s + (−4.49e3 + 7.78e3i)31-s + 1.33e3·35-s − 6.00e3·37-s + (−5.37e3 + 9.30e3i)41-s + (−5.02e3 − 8.70e3i)43-s + (1.17e4 + 2.03e4i)47-s + ⋯ |
| L(s) = 1 | + (−0.0873 + 0.151i)5-s + (−0.527 − 0.912i)7-s + (0.813 + 1.40i)11-s + (−0.205 + 0.356i)13-s − 0.209·17-s − 1.11·19-s + (−0.326 + 0.564i)23-s + (0.484 + 0.839i)25-s + (0.468 + 0.812i)29-s + (−0.839 + 1.45i)31-s + 0.184·35-s − 0.720·37-s + (−0.499 + 0.864i)41-s + (−0.414 − 0.717i)43-s + (0.775 + 1.34i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.571807 + 0.831916i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.571807 + 0.831916i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (4.88 - 8.45i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (68.3 + 118. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-326. - 565. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (125. - 216. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 249.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.75e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (827. - 1.43e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-2.12e3 - 3.67e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (4.49e3 - 7.78e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 6.00e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (5.37e3 - 9.30e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (5.02e3 + 8.70e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.17e4 - 2.03e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 9.41e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-2.20e4 + 3.82e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.12e4 - 1.94e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.80e4 + 3.11e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 7.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (1.37e4 + 2.38e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (3.24e4 + 5.61e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 3.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (8.05e3 + 1.39e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−12.98344022465876466591283342010, −12.17170334945121295646301599198, −10.85929223596314842171622738337, −9.935471737481973468424735594929, −8.891425190804757970747453945262, −7.24360578887385279231446473004, −6.66463133888318735822297035431, −4.76258887522911509754484539289, −3.59087565741181944355999668479, −1.64418581530981571673711098479,
0.37801632988307379797940395377, 2.47599770210853296893627410271, 3.96878384367455217808690191139, 5.69041805596001822141916876797, 6.55803806683671538931144961727, 8.332486132836969501972482564456, 8.988605172340460132002030937548, 10.31632073179217040841054791435, 11.51036742326885948466366051980, 12.39424746005806104815727381912
