| L(s) = 1 | + (2.27 + 3.94i)2-s + (−124. − 71.8i)3-s + (117. − 203. i)4-s + (−163. + 94.1i)5-s − 654. i·6-s + (−2.34e3 − 512. i)7-s + 2.23e3·8-s + (7.03e3 + 1.21e4i)9-s + (−743. − 428. i)10-s + (9.83e3 − 1.70e4i)11-s + (−2.92e4 + 1.68e4i)12-s − 1.40e4i·13-s + (−3.32e3 − 1.04e4i)14-s + 2.70e4·15-s + (−2.50e4 − 4.33e4i)16-s + (−1.47e4 − 8.53e3i)17-s + ⋯ |
| L(s) = 1 | + (0.142 + 0.246i)2-s + (−1.53 − 0.886i)3-s + (0.459 − 0.795i)4-s + (−0.260 + 0.150i)5-s − 0.505i·6-s + (−0.976 − 0.213i)7-s + 0.546·8-s + (1.07 + 1.85i)9-s + (−0.0743 − 0.0428i)10-s + (0.671 − 1.16i)11-s + (−1.41 + 0.814i)12-s − 0.492i·13-s + (−0.0864 − 0.271i)14-s + 0.534·15-s + (−0.381 − 0.660i)16-s + (−0.177 − 0.102i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.339863 - 0.653769i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.339863 - 0.653769i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.34e3 + 512. i)T \) |
| good | 2 | \( 1 + (-2.27 - 3.94i)T + (-128 + 221. i)T^{2} \) |
| 3 | \( 1 + (124. + 71.8i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (163. - 94.1i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-9.83e3 + 1.70e4i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 1.40e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (1.47e4 + 8.53e3i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-2.41e4 + 1.39e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (7.79e4 + 1.35e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 6.40e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-1.52e5 - 8.81e4i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-5.63e5 - 9.76e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 4.38e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.92e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-4.92e6 + 2.84e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (7.35e5 - 1.27e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.68e7 - 9.72e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (8.65e6 - 4.99e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-9.37e6 + 1.62e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.02e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.20e7 - 1.27e7i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.65e7 + 4.59e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 3.78e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (2.19e7 - 1.26e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 6.94e7iT - 7.83e15T^{2} \) |
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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
−19.60179732335998433091305088018, −18.67642854054927945341385095219, −16.94149657494037985882813313817, −15.86325272032766744657230411454, −13.56271722016961889222937885773, −11.84028768229382519083305835061, −10.55578264641755270443778433536, −6.89393598954348580549464949976, −5.80033052623248034873834521742, −0.67676561796902281742861602772,
4.20064513128064730989922244976, 6.58892601761707451401677533557, 9.860932040728076438242941966186, 11.60404480103773333895903610017, 12.48149844109530399520855774994, 15.63953601143289221608695658785, 16.53118179680374709501071586944, 17.65044362388993124445168086463, 19.97923414068460462965518615988, 21.53396571822451467953449489549
