L(s) = 1 | + (0.214 + 0.370i)3-s + (0.323 − 0.561i)5-s + (−2.03 + 3.52i)7-s + (1.40 − 2.43i)9-s + (−0.217 − 0.125i)11-s + (−1.64 − 2.85i)13-s + 0.277·15-s + 1.82i·17-s + (5.79 + 3.34i)19-s − 1.74·21-s + (3.50 + 2.02i)23-s + (2.29 + 3.96i)25-s + 2.49·27-s + (3.46 + 2.00i)29-s + (−1.92 − 1.10i)31-s + ⋯ |
L(s) = 1 | + (0.123 + 0.214i)3-s + (0.144 − 0.250i)5-s + (−0.770 + 1.33i)7-s + (0.469 − 0.813i)9-s + (−0.0656 − 0.0379i)11-s + (−0.456 − 0.790i)13-s + 0.0716·15-s + 0.441i·17-s + (1.32 + 0.767i)19-s − 0.380·21-s + (0.731 + 0.422i)23-s + (0.458 + 0.793i)25-s + 0.479·27-s + (0.644 + 0.371i)29-s + (−0.344 − 0.199i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.586927278\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.586927278\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 79 | \( 1 + (-5.88 - 6.66i)T \) |
good | 3 | \( 1 + (-0.214 - 0.370i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.323 + 0.561i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (2.03 - 3.52i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.217 + 0.125i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.64 + 2.85i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.82iT - 17T^{2} \) |
| 19 | \( 1 + (-5.79 - 3.34i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.50 - 2.02i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.46 - 2.00i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.92 + 1.10i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.78 - 1.02i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 11.9iT - 41T^{2} \) |
| 43 | \( 1 + (-1.51 - 2.62i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.32 - 2.28i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.27 + 2.46i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.25 - 7.37i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 0.126iT - 61T^{2} \) |
| 67 | \( 1 - 1.30iT - 67T^{2} \) |
| 71 | \( 1 + 8.95T + 71T^{2} \) |
| 73 | \( 1 + (-3.80 + 6.58i)T + (-36.5 - 63.2i)T^{2} \) |
| 83 | \( 1 + (1.06 + 0.612i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 1.93T + 97T^{2} \) |
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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
−9.584201810568454233465974387237, −9.254126095450976199902694685957, −8.338157113095563268886429809073, −7.38019192264124594758327051640, −6.37135453283704564658803046391, −5.63712816644694191797240708860, −4.87376549852640225905657377274, −3.41505938750613945293039285905, −2.90307593572397710122096330484, −1.28666796503481677532339741664,
0.74244945034836664798775942308, 2.26648295932277482949464112808, 3.32333708430433292177304425636, 4.41646684720133498346588946182, 5.15746496656699613033703588843, 6.60349385534604359089609555532, 7.10118876187836806262059491502, 7.59911574800252258791262727296, 8.836798868760504892414614283643, 9.662161843661098275201090899624