L(s) = 1 | + (−0.710 + 1.22i)2-s + (1.09 + 1.09i)3-s + (−0.991 − 1.73i)4-s + (−2.11 + 0.560i)6-s + 0.973·7-s + (2.82 + 0.0202i)8-s − 0.616i·9-s + (1.40 + 1.40i)11-s + (0.813 − 2.97i)12-s + (4.60 + 4.60i)13-s + (−0.691 + 1.19i)14-s + (−2.03 + 3.44i)16-s + 0.490i·17-s + (0.754 + 0.438i)18-s + (−4.54 + 4.54i)19-s + ⋯ |
L(s) = 1 | + (−0.502 + 0.864i)2-s + (0.630 + 0.630i)3-s + (−0.495 − 0.868i)4-s + (−0.861 + 0.228i)6-s + 0.368·7-s + (0.999 + 0.00714i)8-s − 0.205i·9-s + (0.424 + 0.424i)11-s + (0.234 − 0.859i)12-s + (1.27 + 1.27i)13-s + (−0.184 + 0.318i)14-s + (−0.508 + 0.861i)16-s + 0.118i·17-s + (0.177 + 0.103i)18-s + (−1.04 + 1.04i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.858541 + 1.03025i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.858541 + 1.03025i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.710 - 1.22i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.09 - 1.09i)T + 3iT^{2} \) |
| 7 | \( 1 - 0.973T + 7T^{2} \) |
| 11 | \( 1 + (-1.40 - 1.40i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.60 - 4.60i)T + 13iT^{2} \) |
| 17 | \( 1 - 0.490iT - 17T^{2} \) |
| 19 | \( 1 + (4.54 - 4.54i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.94T + 23T^{2} \) |
| 29 | \( 1 + (-3.74 + 3.74i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.29T + 31T^{2} \) |
| 37 | \( 1 + (4.55 - 4.55i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (1.79 - 1.79i)T - 43iT^{2} \) |
| 47 | \( 1 - 10.0iT - 47T^{2} \) |
| 53 | \( 1 + (-5.61 + 5.61i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.44 + 8.44i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.01 + 3.01i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.897iT - 71T^{2} \) |
| 73 | \( 1 + 9.71T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + (0.815 + 0.815i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.12iT - 89T^{2} \) |
| 97 | \( 1 + 7.54iT - 97T^{2} \) |
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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
−11.29004082795980394063018669300, −10.24960558137987380343229211202, −9.545206668266864098368341045105, −8.624784316492124592914150100145, −8.202725756533915165150074045658, −6.72685000690423340840624807724, −6.11562312499777075021878573265, −4.53248748115910017846677775452, −3.85844149846498808693410811657, −1.70115842802557064105826974975,
1.15028691645973194058619594297, 2.50490249927942166459291898534, 3.55269388685797793419705357396, 4.93257606466009361890962816633, 6.49827399102592194612315666409, 7.70915410562306181151991482685, 8.457605567618762978850856921289, 8.893047283455226291826239942309, 10.37328644278565490156072863518, 10.88036231878435304213856135084