| L(s) = 1 | + (−4.5 − 7.79i)9-s + (−3 + 5.19i)11-s + (9 + 15.5i)23-s + (−12.5 + 21.6i)25-s − 54·29-s + (19 + 32.9i)37-s − 58·43-s + (3 − 5.19i)53-s + (−59 + 102. i)67-s − 114·71-s + (−47 − 81.4i)79-s + (−40.5 + 70.1i)81-s + 54·99-s + (93 + 161. i)107-s + (−53 + 91.7i)109-s + ⋯ |
| L(s) = 1 | + (−0.5 − 0.866i)9-s + (−0.272 + 0.472i)11-s + (0.391 + 0.677i)23-s + (−0.5 + 0.866i)25-s − 1.86·29-s + (0.513 + 0.889i)37-s − 1.34·43-s + (0.0566 − 0.0980i)53-s + (−0.880 + 1.52i)67-s − 1.60·71-s + (−0.594 − 1.03i)79-s + (−0.5 + 0.866i)81-s + 0.545·99-s + (0.869 + 1.50i)107-s + (−0.486 + 0.842i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4181333147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4181333147\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-9 - 15.5i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 54T + 841T^{2} \) |
| 31 | \( 1 + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-19 - 32.9i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 58T + 1.84e3T^{2} \) |
| 47 | \( 1 + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (59 - 102. i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 114T + 5.04e3T^{2} \) |
| 73 | \( 1 + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (47 + 81.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−10.35315736769353695869385861903, −9.505799373449186855538338750620, −8.901312000742292124836074615787, −7.79914406684601639039821636583, −7.03819254733292251795181515570, −5.99452478341161084892939450672, −5.19212558268515825653671177405, −3.93849458255371702142942639794, −3.00695691929774471893266182892, −1.56067324444711608398521567525,
0.13597940594485749797925600142, 1.95796298803177041604716945598, 3.04995067127091513370221035668, 4.29275047522763052850282234577, 5.34420800088296405865570050431, 6.09001907185173219185400072540, 7.27237035640786299273039636240, 8.072255464685806775833847449375, 8.802227709768250425766156382581, 9.781670440993358818733522345636
