L(s) = 1 | − i·2-s − i·3-s − 4-s + (−1.67 − 1.48i)5-s − 6-s + 2.96i·7-s + i·8-s − 9-s + (−1.48 + 1.67i)10-s − 3.35·11-s + i·12-s + 4.96i·13-s + 2.96·14-s + (−1.48 + 1.67i)15-s + 16-s + 1.35i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.749 − 0.662i)5-s − 0.408·6-s + 1.11i·7-s + 0.353i·8-s − 0.333·9-s + (−0.468 + 0.529i)10-s − 1.01·11-s + 0.288i·12-s + 1.37i·13-s + 0.791·14-s + (−0.382 + 0.432i)15-s + 0.250·16-s + 0.327i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.537578 + 0.242251i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.537578 + 0.242251i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.67 + 1.48i)T \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 - 2.96iT - 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 - 4.96iT - 13T^{2} \) |
| 17 | \( 1 - 1.35iT - 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 29 | \( 1 + 7.73T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 7.61iT - 37T^{2} \) |
| 41 | \( 1 - 4.70T + 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 3.22iT - 47T^{2} \) |
| 53 | \( 1 - 6.96iT - 53T^{2} \) |
| 59 | \( 1 + 1.22T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 7.61iT - 67T^{2} \) |
| 71 | \( 1 + 2.18T + 71T^{2} \) |
| 73 | \( 1 - 9.92iT - 73T^{2} \) |
| 79 | \( 1 - 4.12T + 79T^{2} \) |
| 83 | \( 1 - 6.38iT - 83T^{2} \) |
| 89 | \( 1 + 9.92T + 89T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
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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
−10.97847797193717029031664038703, −9.329395430959824711992303907421, −9.159980262017052137001948087874, −7.999080136219275650084962508307, −7.39010595834352775976338951518, −5.88160987938398970056839817903, −5.11996696825022873012102947786, −3.97279167040471620375744686712, −2.71571553421871008147453122111, −1.57517351925560974559097061736,
0.31488607278915811316398700563, 3.04720196881521653527097991728, 3.80224340879444563616649041099, 4.94530519209386311746993164876, 5.76392117959341146464651159109, 7.17981994587165152437397551930, 7.56120688088907304674627686932, 8.338450402531031311380631667170, 9.611605931127727491480621860268, 10.44482545388560738436856991168