L(s) = 1 | + (−0.702 − 1.22i)2-s + (1.63 − 1.63i)3-s + (−1.01 + 1.72i)4-s + (−1.85 − 1.24i)5-s + (−3.15 − 0.856i)6-s + (1.80 − 1.80i)7-s + (2.82 + 0.0296i)8-s − 2.34i·9-s + (−0.229 + 3.15i)10-s − 2.86·11-s + (1.16 + 4.47i)12-s + (−1.91 − 3.05i)13-s + (−3.47 − 0.944i)14-s + (−5.07 + 0.989i)15-s + (−1.95 − 3.49i)16-s + (3.05 − 3.05i)17-s + ⋯ |
L(s) = 1 | + (−0.496 − 0.867i)2-s + (0.943 − 0.943i)3-s + (−0.506 + 0.862i)4-s + (−0.829 − 0.558i)5-s + (−1.28 − 0.349i)6-s + (0.681 − 0.681i)7-s + (0.999 + 0.0104i)8-s − 0.780i·9-s + (−0.0726 + 0.997i)10-s − 0.864·11-s + (0.336 + 1.29i)12-s + (−0.531 − 0.847i)13-s + (−0.929 − 0.252i)14-s + (−1.30 + 0.255i)15-s + (−0.487 − 0.872i)16-s + (0.739 − 0.739i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.243091 - 1.05221i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.243091 - 1.05221i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.702 + 1.22i)T \) |
| 5 | \( 1 + (1.85 + 1.24i)T \) |
| 13 | \( 1 + (1.91 + 3.05i)T \) |
good | 3 | \( 1 + (-1.63 + 1.63i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1.80 + 1.80i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 17 | \( 1 + (-3.05 + 3.05i)T - 17iT^{2} \) |
| 19 | \( 1 - 5.05iT - 19T^{2} \) |
| 23 | \( 1 + (2.42 - 2.42i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.79iT - 29T^{2} \) |
| 31 | \( 1 - 7.49T + 31T^{2} \) |
| 37 | \( 1 + (-2.43 - 2.43i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.73iT - 41T^{2} \) |
| 43 | \( 1 + (1.36 - 1.36i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.11 + 7.11i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.33 + 2.33i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.71iT - 59T^{2} \) |
| 61 | \( 1 - 9.15T + 61T^{2} \) |
| 67 | \( 1 + (1.42 - 1.42i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.80T + 71T^{2} \) |
| 73 | \( 1 + (-5.53 + 5.53i)T - 73iT^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + (-3.28 - 3.28i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.690T + 89T^{2} \) |
| 97 | \( 1 + (-0.611 - 0.611i)T + 97iT^{2} \) |
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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
−11.84609918736893572193885775533, −10.59378634879977871268270106679, −9.709852758250580993943551097453, −8.274610239960415420849790243288, −7.916336792542197831385027082143, −7.43399070121323644112945182568, −5.06579648706614356514424814232, −3.75004027093847040417281419271, −2.50247258303835335516860294644, −0.962949218932236580241678889107,
2.62643286112913370868709588654, 4.20498929568939586091839162677, 5.09386478664477996570842959744, 6.65199025609995757847826596522, 7.85661694341215072344081300819, 8.415452995221247643703962277690, 9.307346440124177146851343662650, 10.27331679622084179957438848442, 11.08334428710807524742728842929, 12.33023940660766315776238639234