| L(s) = 1 | + (−2.76 + 1.14i)2-s + (−1.69 − 0.337i)3-s + (3.50 − 3.50i)4-s + (−1.01 − 1.52i)5-s + (5.08 − 1.01i)6-s + (5.94 − 8.89i)7-s + (−1.09 + 2.65i)8-s + (2.77 + 1.14i)9-s + (4.56 + 3.04i)10-s + (−1.10 − 5.55i)11-s + (−7.14 + 4.77i)12-s + (−7.64 − 7.64i)13-s + (−6.24 + 31.4i)14-s + (1.21 + 2.93i)15-s + 11.2i·16-s + (−13.7 − 9.96i)17-s + ⋯ |
| L(s) = 1 | + (−1.38 + 0.572i)2-s + (−0.566 − 0.112i)3-s + (0.876 − 0.876i)4-s + (−0.203 − 0.304i)5-s + (0.847 − 0.168i)6-s + (0.849 − 1.27i)7-s + (−0.137 + 0.331i)8-s + (0.307 + 0.127i)9-s + (0.456 + 0.304i)10-s + (−0.100 − 0.504i)11-s + (−0.595 + 0.397i)12-s + (−0.588 − 0.588i)13-s + (−0.446 + 2.24i)14-s + (0.0809 + 0.195i)15-s + 0.702i·16-s + (−0.809 − 0.586i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.382754 - 0.216490i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.382754 - 0.216490i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.69 + 0.337i)T \) |
| 17 | \( 1 + (13.7 + 9.96i)T \) |
| good | 2 | \( 1 + (2.76 - 1.14i)T + (2.82 - 2.82i)T^{2} \) |
| 5 | \( 1 + (1.01 + 1.52i)T + (-9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (-5.94 + 8.89i)T + (-18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (1.10 + 5.55i)T + (-111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (7.64 + 7.64i)T + 169iT^{2} \) |
| 19 | \( 1 + (10.0 - 4.16i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-35.7 + 7.10i)T + (488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-16.5 + 11.0i)T + (321. - 776. i)T^{2} \) |
| 31 | \( 1 + (8.38 - 42.1i)T + (-887. - 367. i)T^{2} \) |
| 37 | \( 1 + (40.0 + 7.95i)T + (1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (0.900 - 1.34i)T + (-643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-47.3 - 19.6i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-22.5 - 22.5i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (76.5 - 31.7i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (7.23 - 17.4i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-89.1 - 59.5i)T + (1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 + 111. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-99.1 - 19.7i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (5.05 + 7.56i)T + (-2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (15.5 + 78.0i)T + (-5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-25.9 - 62.7i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-26.7 + 26.7i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (10.4 - 6.97i)T + (3.60e3 - 8.69e3i)T^{2} \) |
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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
−15.59134720918813602999229569059, −14.13520217742075835473984404089, −12.70553717702383589965312590300, −11.00907992830944538843061100083, −10.42775045978163418829348215366, −8.834168236714763905667054962612, −7.72333006788937967034691001182, −6.75212457796353878464095739903, −4.72772192462511405832695120843, −0.76614042355861564365217977373,
2.12755714190587336040065682039, 5.06274059730198870503003505518, 7.06456800998173546246818491872, 8.534032065117740939004019818336, 9.451028023346228016477171872228, 10.89513154528967482494494061729, 11.48764775424148362575192556347, 12.60635948337160156786491764830, 14.71130726999879082015312446994, 15.58025323762963122639144422042
