L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.24 + 2.15i)5-s + 0.999·6-s + (2.47 + 0.929i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.24 + 2.15i)10-s + (1.73 + 2.99i)11-s + (0.499 − 0.866i)12-s − 4.98·13-s + (2.04 − 1.68i)14-s − 2.49·15-s + (−0.5 + 0.866i)16-s + (−1.12 − 1.94i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.557 + 0.965i)5-s + 0.408·6-s + (0.936 + 0.351i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.394 + 0.682i)10-s + (0.521 + 0.903i)11-s + (0.144 − 0.249i)12-s − 1.38·13-s + (0.546 − 0.449i)14-s − 0.643·15-s + (−0.125 + 0.216i)16-s + (−0.272 − 0.471i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30985 + 0.970475i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30985 + 0.970475i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.47 - 0.929i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (1.24 - 2.15i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.73 - 2.99i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.98T + 13T^{2} \) |
| 17 | \( 1 + (1.12 + 1.94i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (3.00 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.51T + 29T^{2} \) |
| 31 | \( 1 + (-3.64 - 6.30i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.58 + 7.93i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 - 9.54T + 43T^{2} \) |
| 47 | \( 1 + (2.68 - 4.64i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.93 - 5.08i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.33 - 7.51i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.18 - 3.78i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.57 + 13.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + (7.21 + 12.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.54 + 6.13i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.59T + 83T^{2} \) |
| 89 | \( 1 + (3.76 - 6.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−10.61868503713108248958395180511, −9.636242896508000792370884205760, −9.087058144381190762064201026605, −7.64556222266778381876118309330, −7.29678992859357501474182914781, −5.81304363847087432844234914459, −4.74687970818270516791400085736, −4.07649851679416914230629760012, −2.88288993892061144680541578947, −1.97032111655133572341905052556,
0.70217313964319743459085512356, 2.37704828708127324626432141570, 4.01681839210996036307627232106, 4.59531648617437197206587770056, 5.64995413672672967147243563885, 6.69269134457663136743511192300, 7.69837552029603347002841277220, 8.220651692928996736194043636240, 8.837436259394706769851973924131, 9.913856301923551882143402774145