L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.22 − 1.22i)3-s + 1.00i·4-s − 1.73·6-s + (−1.50 + 1.50i)7-s + (0.707 − 0.707i)8-s − 5.03i·11-s + (1.22 + 1.22i)12-s + (−3.34 + 3.34i)13-s + 2.12·14-s − 1.00·16-s + (−5.06 + 5.06i)17-s − 5.03·19-s + 3.68i·21-s + (−3.56 + 3.56i)22-s + (−2.91 − 3.80i)23-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.707 − 0.707i)3-s + 0.500i·4-s − 0.707·6-s + (−0.569 + 0.569i)7-s + (0.250 − 0.250i)8-s − 1.51i·11-s + (0.353 + 0.353i)12-s + (−0.928 + 0.928i)13-s + 0.569·14-s − 0.250·16-s + (−1.22 + 1.22i)17-s − 1.15·19-s + 0.804i·21-s + (−0.759 + 0.759i)22-s + (−0.607 − 0.794i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09014913268\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09014913268\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (2.91 + 3.80i)T \) |
good | 3 | \( 1 + (-1.22 + 1.22i)T - 3iT^{2} \) |
| 7 | \( 1 + (1.50 - 1.50i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.03iT - 11T^{2} \) |
| 13 | \( 1 + (3.34 - 3.34i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.06 - 5.06i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.03T + 19T^{2} \) |
| 29 | \( 1 + 5.46iT - 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 + (4.11 - 4.11i)T - 37iT^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + (-1.10 - 1.10i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.34 + 3.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (9.73 + 9.73i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.535iT - 59T^{2} \) |
| 61 | \( 1 - 3.68iT - 61T^{2} \) |
| 67 | \( 1 + (-2.05 + 2.05i)T - 67iT^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + (-0.568 + 0.568i)T - 73iT^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + (1.10 - 1.10i)T - 97iT^{2} \) |
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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
−10.04621844918524141954247896164, −9.009348480827511830611252190602, −8.433565123291351949881927424230, −8.069286636419673865796616368874, −6.58591990118403917413252528408, −6.38551559127811720785036203786, −4.74462873688725444258169524708, −3.59610393444627790717437682372, −2.50723911342600226815870484252, −1.89905913669511466711614726012,
0.03800909347313345600066192224, 2.12294245595762346096391460851, 3.24732940249303505327312108653, 4.43055131021268097978890082067, 5.01366736487567886572519837615, 6.51250331935515237201602857715, 7.08915092261120935273242631950, 7.88223583698667773919939208416, 8.906957086093588950535979929020, 9.496926851503714260599368060746