L(s) = 1 | + (−0.874 − 1.11i)2-s + 2.41i·3-s + (−0.470 + 1.94i)4-s + i·5-s + (2.68 − 2.11i)6-s − 7-s + (2.57 − 1.17i)8-s − 2.82·9-s + (1.11 − 0.874i)10-s + 1.66i·11-s + (−4.69 − 1.13i)12-s − 0.143i·13-s + (0.874 + 1.11i)14-s − 2.41·15-s + (−3.55 − 1.82i)16-s − 6.03·17-s + ⋯ |
L(s) = 1 | + (−0.618 − 0.785i)2-s + 1.39i·3-s + (−0.235 + 0.971i)4-s + 0.447i·5-s + (1.09 − 0.861i)6-s − 0.377·7-s + (0.909 − 0.416i)8-s − 0.942·9-s + (0.351 − 0.276i)10-s + 0.503i·11-s + (−1.35 − 0.327i)12-s − 0.0397i·13-s + (0.233 + 0.297i)14-s − 0.623·15-s + (−0.889 − 0.457i)16-s − 1.46·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.372542 + 0.580289i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.372542 + 0.580289i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.874 + 1.11i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 2.41iT - 3T^{2} \) |
| 11 | \( 1 - 1.66iT - 11T^{2} \) |
| 13 | \( 1 + 0.143iT - 13T^{2} \) |
| 17 | \( 1 + 6.03T + 17T^{2} \) |
| 19 | \( 1 - 6.64iT - 19T^{2} \) |
| 23 | \( 1 + 5.30T + 23T^{2} \) |
| 29 | \( 1 + 3.67iT - 29T^{2} \) |
| 31 | \( 1 - 6.80T + 31T^{2} \) |
| 37 | \( 1 + 4.35iT - 37T^{2} \) |
| 41 | \( 1 - 8.64T + 41T^{2} \) |
| 43 | \( 1 - 8.80iT - 43T^{2} \) |
| 47 | \( 1 - 7.13T + 47T^{2} \) |
| 53 | \( 1 + 5.11iT - 53T^{2} \) |
| 59 | \( 1 + 8.40iT - 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 9.11iT - 67T^{2} \) |
| 71 | \( 1 + 3.89T + 71T^{2} \) |
| 73 | \( 1 + 1.89T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 1.77iT - 83T^{2} \) |
| 89 | \( 1 - 7.13T + 89T^{2} \) |
| 97 | \( 1 - 5.23T + 97T^{2} \) |
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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
−11.82775061040420517768309485349, −10.93814646652475343942278548543, −10.13824033688975631967425492554, −9.706931084196016898339397456999, −8.711315576090193601434451570699, −7.60272749687795178261256718097, −6.16402060591168302121437417908, −4.45698306896100415107889455091, −3.79776384714988445567939271618, −2.40258374641210483121512696510,
0.63019074116518646852955770486, 2.25161435901464933315574593362, 4.61688458028020883038745895956, 6.03979757859934310494658150267, 6.70400360240446792828503573104, 7.57760748185308139767289409134, 8.567870478037145981942080561982, 9.218077997264062937358533848197, 10.57920038999316246637335338856, 11.60510480372104757347478054314