L(s) = 1 | + (0.856 + 3.19i)3-s + (−1.51 − 1.64i)5-s + (−0.802 + 2.52i)7-s + (−6.87 + 3.97i)9-s + (−1.05 + 1.82i)11-s + (1.20 − 1.20i)13-s + (3.97 − 6.23i)15-s + (3.17 − 0.850i)17-s + (2.36 + 4.09i)19-s + (−8.74 − 0.406i)21-s + (1.07 − 4.00i)23-s + (−0.436 + 4.98i)25-s + (−11.5 − 11.5i)27-s + 3.65i·29-s + (−1.63 − 0.946i)31-s + ⋯ |
L(s) = 1 | + (0.494 + 1.84i)3-s + (−0.675 − 0.737i)5-s + (−0.303 + 0.952i)7-s + (−2.29 + 1.32i)9-s + (−0.317 + 0.550i)11-s + (0.334 − 0.334i)13-s + (1.02 − 1.61i)15-s + (0.770 − 0.206i)17-s + (0.542 + 0.939i)19-s + (−1.90 − 0.0886i)21-s + (0.223 − 0.835i)23-s + (−0.0872 + 0.996i)25-s + (−2.22 − 2.22i)27-s + 0.679i·29-s + (−0.294 − 0.169i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.437801 + 1.06696i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.437801 + 1.06696i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.51 + 1.64i)T \) |
| 7 | \( 1 + (0.802 - 2.52i)T \) |
good | 3 | \( 1 + (-0.856 - 3.19i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (1.05 - 1.82i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.20 + 1.20i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.17 + 0.850i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.36 - 4.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.07 + 4.00i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 3.65iT - 29T^{2} \) |
| 31 | \( 1 + (1.63 + 0.946i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.91 - 2.65i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 0.826iT - 41T^{2} \) |
| 43 | \( 1 + (-4.70 - 4.70i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.04 + 3.90i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.40 + 0.645i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.15 + 2.00i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.44 + 0.837i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.28 - 12.2i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + (3.37 + 12.6i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (8.29 - 4.78i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.47 + 3.47i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.86 + 4.96i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.02 - 1.02i)T + 97iT^{2} \) |
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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
−12.07755557728397327217049699000, −11.16286160981384892875068326536, −10.08778600835802001814016920045, −9.429863961169736770210567174718, −8.589814869858623662455130182518, −7.85693800645980229100886956393, −5.71668647786113159896501589516, −4.96120250545378231757311934736, −3.90950609417636629050493402559, −2.86377554257275970934460538850,
0.861798242784509726565305780668, 2.69418908078590692294170694694, 3.69939298789886291254054506065, 5.90730088626218506242979161442, 6.88523616264141312237922942908, 7.52242755660135993181820116971, 8.153549151264825320616761918965, 9.435254475695859936923170251627, 10.95333976226138042331046364577, 11.55024639209073753182763520298