L(s) = 1 | + (−0.409 + 0.409i)3-s + (2.54 + 0.738i)7-s + 2.66i·9-s − 3.54·11-s + (−2.95 + 2.95i)13-s + (−5.59 − 5.59i)17-s − 3.59·19-s + (−1.34 + 0.737i)21-s + (0.0472 + 0.0472i)23-s + (−2.31 − 2.31i)27-s − 5.34i·29-s + 10.3i·31-s + (1.45 − 1.45i)33-s + (7.80 − 7.80i)37-s − 2.41i·39-s + ⋯ |
L(s) = 1 | + (−0.236 + 0.236i)3-s + (0.960 + 0.278i)7-s + 0.888i·9-s − 1.07·11-s + (−0.818 + 0.818i)13-s + (−1.35 − 1.35i)17-s − 0.824·19-s + (−0.292 + 0.160i)21-s + (0.00986 + 0.00986i)23-s + (−0.446 − 0.446i)27-s − 0.993i·29-s + 1.86i·31-s + (0.252 − 0.252i)33-s + (1.28 − 1.28i)37-s − 0.386i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3352517557\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3352517557\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.54 - 0.738i)T \) |
good | 3 | \( 1 + (0.409 - 0.409i)T - 3iT^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 + (2.95 - 2.95i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.59 + 5.59i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 23 | \( 1 + (-0.0472 - 0.0472i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.34iT - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + (-7.80 + 7.80i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.63iT - 41T^{2} \) |
| 43 | \( 1 + (6.93 + 6.93i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.44 + 3.44i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.646 - 0.646i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.22T + 59T^{2} \) |
| 61 | \( 1 - 3.51iT - 61T^{2} \) |
| 67 | \( 1 + (1.70 - 1.70i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.54T + 71T^{2} \) |
| 73 | \( 1 + (2.43 - 2.43i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.72iT - 79T^{2} \) |
| 83 | \( 1 + (2.04 - 2.04i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.18T + 89T^{2} \) |
| 97 | \( 1 + (6.23 + 6.23i)T + 97iT^{2} \) |
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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
−10.07254017252179908094877198777, −9.105316759411328639112331649794, −8.371362377110089119352871340603, −7.56343887856968364537744639415, −6.85453360914377599269995216443, −5.61551077563664771693571013832, −4.75953625033441796008761151920, −4.50302591672537156608230146364, −2.62728026584577022442830145705, −2.02287985380198908771155862564,
0.12830617763390862700091743132, 1.70646866022045643206245659284, 2.82738591720294852370819838123, 4.15948756008448552665345938417, 4.87919208035860505786453473664, 5.91498035409328392722994427708, 6.62521753601709050847143894490, 7.72867922716256434678667720853, 8.151868456921288424295754761626, 9.089147253888561166895176665031