| L(s) = 1 | + (−0.630 − 1.23i)3-s + (2.13 − 0.668i)5-s + (−1.04 − 1.04i)7-s + (0.630 − 0.868i)9-s + (−0.0991 − 0.136i)11-s + (−0.0572 − 0.361i)13-s + (−2.17 − 2.21i)15-s + (−1.20 − 0.614i)17-s + (0.488 + 1.50i)19-s + (−0.635 + 1.95i)21-s + (1.02 − 6.45i)23-s + (4.10 − 2.85i)25-s + (−5.58 − 0.884i)27-s + (−3.38 − 1.09i)29-s + (3.16 − 1.02i)31-s + ⋯ |
| L(s) = 1 | + (−0.363 − 0.714i)3-s + (0.954 − 0.299i)5-s + (−0.396 − 0.396i)7-s + (0.210 − 0.289i)9-s + (−0.0298 − 0.0411i)11-s + (−0.0158 − 0.100i)13-s + (−0.560 − 0.572i)15-s + (−0.292 − 0.149i)17-s + (0.111 + 0.344i)19-s + (−0.138 + 0.426i)21-s + (0.213 − 1.34i)23-s + (0.821 − 0.570i)25-s + (−1.07 − 0.170i)27-s + (−0.628 − 0.204i)29-s + (0.568 − 0.184i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-1.90e-5 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-1.90e-5 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.935912 - 0.935930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.935912 - 0.935930i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.13 + 0.668i)T \) |
| good | 3 | \( 1 + (0.630 + 1.23i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (1.04 + 1.04i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.0991 + 0.136i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.0572 + 0.361i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (1.20 + 0.614i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.488 - 1.50i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.02 + 6.45i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (3.38 + 1.09i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.16 + 1.02i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (3.57 - 0.566i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-6.74 - 4.89i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.31 + 4.31i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.72 - 1.89i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (2.34 - 1.19i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-10.3 - 7.52i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (7.31 - 5.31i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (5.12 - 10.0i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-15.4 - 5.02i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-14.1 - 2.23i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (4.87 - 15.0i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-11.4 - 5.85i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-2.77 - 3.82i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (5.33 + 10.4i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.02758939671431458313759129854, −10.07466706563591299729638415648, −9.357650923225285935413048141289, −8.262315675703684903101602205201, −7.02364795353807184363246415807, −6.38932531650397112698946698360, −5.45416806424891761116911186426, −4.10827271265633567582044795005, −2.44278264860091149958667575414, −0.959375906666444479602816549010,
2.00020943284876486563931319416, 3.42197325228882822184131846156, 4.84278485695295262040373276546, 5.64583043123360909085765919554, 6.60592559340031675465976772818, 7.73995592885622184026277381080, 9.271153943459599151840645407551, 9.554213825494681129989384657521, 10.64403088404666956129427996859, 11.16672179362240804777206997877
