L(s) = 1 | + (−0.470 − 1.33i)2-s − 0.528i·3-s + (−1.55 + 1.25i)4-s + 5-s + (−0.704 + 0.248i)6-s + (2.17 + 1.50i)7-s + (2.40 + 1.48i)8-s + 2.72·9-s + (−0.470 − 1.33i)10-s + 3.04·11-s + (0.663 + 0.822i)12-s − 4.75·13-s + (0.985 − 3.60i)14-s − 0.528i·15-s + (0.844 − 3.90i)16-s − 6.78i·17-s + ⋯ |
L(s) = 1 | + (−0.333 − 0.942i)2-s − 0.304i·3-s + (−0.778 + 0.628i)4-s + 0.447·5-s + (−0.287 + 0.101i)6-s + (0.821 + 0.569i)7-s + (0.851 + 0.524i)8-s + 0.907·9-s + (−0.148 − 0.421i)10-s + 0.918·11-s + (0.191 + 0.237i)12-s − 1.31·13-s + (0.263 − 0.964i)14-s − 0.136i·15-s + (0.211 − 0.977i)16-s − 1.64i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03444 - 0.676569i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03444 - 0.676569i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.470 + 1.33i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.17 - 1.50i)T \) |
good | 3 | \( 1 + 0.528iT - 3T^{2} \) |
| 11 | \( 1 - 3.04T + 11T^{2} \) |
| 13 | \( 1 + 4.75T + 13T^{2} \) |
| 17 | \( 1 + 6.78iT - 17T^{2} \) |
| 19 | \( 1 - 0.584iT - 19T^{2} \) |
| 23 | \( 1 - 5.80iT - 23T^{2} \) |
| 29 | \( 1 - 0.185iT - 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 7.04iT - 37T^{2} \) |
| 41 | \( 1 - 3.83iT - 41T^{2} \) |
| 43 | \( 1 + 1.43T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 - 0.535iT - 53T^{2} \) |
| 59 | \( 1 - 1.52iT - 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 9.13T + 67T^{2} \) |
| 71 | \( 1 - 9.68iT - 71T^{2} \) |
| 73 | \( 1 + 12.4iT - 73T^{2} \) |
| 79 | \( 1 - 2.81iT - 79T^{2} \) |
| 83 | \( 1 - 13.4iT - 83T^{2} \) |
| 89 | \( 1 - 3.10iT - 89T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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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
−11.96410451773751381454220535350, −10.81579084857764221444565586736, −9.555047838689482103242439138184, −9.320785008306851606929961935067, −7.85692849158345478356167681084, −7.07284570927092457009192814790, −5.30923863089597553140442344181, −4.36083819002557442810050634637, −2.64005542417613593550426469598, −1.45396513664675929370458057608,
1.54886346216681532228131749993, 4.17625232983370484405802455462, 4.83895923451092101092800841944, 6.25987937149316841481202695136, 7.12567886426798079024758108177, 8.115016154028922498660003756805, 9.114336017085564269223816964964, 10.15161970469342167888532674902, 10.57610788672612320979884196836, 12.14228496337508304518809713701