| L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (−1.93 + 1.11i)5-s + (5.10 + 4.79i)7-s + 2.82·8-s + (2.73 + 1.58i)10-s + (−0.919 + 1.59i)11-s + 5.40i·13-s + (2.26 − 9.63i)14-s + (−2.00 − 3.46i)16-s + (−8.71 − 5.02i)17-s + (−7.96 + 4.59i)19-s − 4.47i·20-s + 2.60·22-s + (−0.460 − 0.797i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.728 + 0.684i)7-s + 0.353·8-s + (0.273 + 0.158i)10-s + (−0.0835 + 0.144i)11-s + 0.415i·13-s + (0.161 − 0.688i)14-s + (−0.125 − 0.216i)16-s + (−0.512 − 0.295i)17-s + (−0.418 + 0.241i)19-s − 0.223i·20-s + 0.118·22-s + (−0.0200 − 0.0346i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6083981754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6083981754\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
| 7 | \( 1 + (-5.10 - 4.79i)T \) |
| good | 11 | \( 1 + (0.919 - 1.59i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 5.40iT - 169T^{2} \) |
| 17 | \( 1 + (8.71 + 5.02i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (7.96 - 4.59i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (0.460 + 0.797i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 12.5T + 841T^{2} \) |
| 31 | \( 1 + (36.1 + 20.8i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (3.64 + 6.30i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 52.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8.12T + 1.84e3T^{2} \) |
| 47 | \( 1 + (29.4 - 16.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (52.0 - 90.1i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (12.5 + 7.26i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-20.5 + 11.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-7.46 + 12.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 17.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (107. + 62.1i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-40.4 - 70.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 154. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (58.9 - 34.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 88.1iT - 9.40e3T^{2} \) |
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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
−10.98415216025210848318696527112, −9.733017870224931747547368111332, −9.010532195136518144165377807754, −8.153628916999270526721094189060, −7.38743726330466048741723446078, −6.18895818901069489393477263104, −4.95642916975452061759492384199, −4.02884020317118029858642148745, −2.71587914308965195007552624963, −1.65427316412905603397328642757,
0.24996681915512855531569785918, 1.75064008421820999431197560904, 3.61418916343563796700818055233, 4.65747891645401251019083329666, 5.52398054877504914658974154648, 6.75672538374100582216267356431, 7.50417580301408260913688204854, 8.328354953402750358252144408526, 8.975209398800079930474998128527, 10.16489037086027595484367743655
