L(s) = 1 | + (−1.49 + 1.49i)2-s + (−1.00 − 1.41i)3-s − 2.49i·4-s + (0.189 + 0.706i)5-s + (3.61 + 0.612i)6-s + (−2.64 − 0.127i)7-s + (0.737 + 0.737i)8-s + (−0.987 + 2.83i)9-s + (−1.34 − 0.774i)10-s + (3.26 − 0.874i)11-s + (−3.51 + 2.49i)12-s + (3.48 − 0.929i)13-s + (4.15 − 3.76i)14-s + (0.807 − 0.975i)15-s + 2.77·16-s + 4.62·17-s + ⋯ |
L(s) = 1 | + (−1.05 + 1.05i)2-s + (−0.579 − 0.815i)3-s − 1.24i·4-s + (0.0846 + 0.315i)5-s + (1.47 + 0.250i)6-s + (−0.998 − 0.0482i)7-s + (0.260 + 0.260i)8-s + (−0.329 + 0.944i)9-s + (−0.424 − 0.245i)10-s + (0.984 − 0.263i)11-s + (−1.01 + 0.721i)12-s + (0.966 − 0.257i)13-s + (1.10 − 1.00i)14-s + (0.208 − 0.251i)15-s + 0.693·16-s + 1.12·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.517496 + 0.260308i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.517496 + 0.260308i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.00 + 1.41i)T \) |
| 7 | \( 1 + (2.64 + 0.127i)T \) |
| 13 | \( 1 + (-3.48 + 0.929i)T \) |
good | 2 | \( 1 + (1.49 - 1.49i)T - 2iT^{2} \) |
| 5 | \( 1 + (-0.189 - 0.706i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-3.26 + 0.874i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 - 4.62T + 17T^{2} \) |
| 19 | \( 1 + (1.69 - 6.31i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 + (-0.0742 + 0.0428i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.75 - 6.53i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-3.78 + 3.78i)T - 37iT^{2} \) |
| 41 | \( 1 + (-11.3 - 3.04i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (7.89 + 4.55i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.11 + 0.297i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.966 - 0.558i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.0904 - 0.0904i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.604 - 1.04i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.33 + 1.96i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.179 + 0.671i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (3.83 + 1.02i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.74 + 13.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.66 - 4.66i)T - 83iT^{2} \) |
| 89 | \( 1 + (8.63 + 8.63i)T + 89iT^{2} \) |
| 97 | \( 1 + (7.29 - 1.95i)T + (84.0 - 48.5i)T^{2} \) |
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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
−12.16912045472024968842769073230, −10.81692772203594846644047238334, −10.03411239105139068777677395249, −8.927037160501292988052617064816, −8.052729353771191189258400937319, −7.06343364835785509363654558170, −6.28258737490604709489746450584, −5.77072929399928430595424712495, −3.45592569926075756020883213865, −1.08618145191779942542608293062,
0.929537771476377443791623371600, 3.01326861732023887735939593272, 4.11855726063650048938138168266, 5.70817762433088073726820917660, 6.79039704446093312303354083947, 8.513825533053364942204815684477, 9.449450901145461785797312113948, 9.613507241989911903974345056676, 10.86442772031085669406372059246, 11.38999340911927180127728657599