L(s) = 1 | + (−1.66 − 1.66i)3-s + (1.34 + 2.27i)7-s + 2.56i·9-s + 1.56·11-s + (−2.60 − 2.60i)13-s + (−2.60 + 2.60i)17-s + 2.64·19-s + (1.56 − 6.04i)21-s + (−0.794 + 0.794i)23-s + (−0.731 + 0.731i)27-s + 6.68i·29-s + 9.43i·31-s + (−2.60 − 2.60i)33-s + (−2.82 − 2.82i)37-s + 8.68i·39-s + ⋯ |
L(s) = 1 | + (−0.962 − 0.962i)3-s + (0.507 + 0.861i)7-s + 0.853i·9-s + 0.470·11-s + (−0.722 − 0.722i)13-s + (−0.631 + 0.631i)17-s + 0.607·19-s + (0.340 − 1.31i)21-s + (−0.165 + 0.165i)23-s + (−0.140 + 0.140i)27-s + 1.24i·29-s + 1.69i·31-s + (−0.453 − 0.453i)33-s + (−0.464 − 0.464i)37-s + 1.39i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9085667004\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9085667004\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.34 - 2.27i)T \) |
good | 3 | \( 1 + (1.66 + 1.66i)T + 3iT^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 + (2.60 + 2.60i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.60 - 2.60i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 + (0.794 - 0.794i)T - 23iT^{2} \) |
| 29 | \( 1 - 6.68iT - 29T^{2} \) |
| 31 | \( 1 - 9.43iT - 31T^{2} \) |
| 37 | \( 1 + (2.82 + 2.82i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.64iT - 41T^{2} \) |
| 43 | \( 1 + (6.45 - 6.45i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.66 - 1.66i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.24 + 7.24i)T - 53iT^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 9.43iT - 61T^{2} \) |
| 67 | \( 1 + (-3.62 - 3.62i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + (-6.67 - 6.67i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.8iT - 79T^{2} \) |
| 83 | \( 1 + (-9.47 - 9.47i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.43T + 89T^{2} \) |
| 97 | \( 1 + (-5.52 + 5.52i)T - 97iT^{2} \) |
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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
−9.663765381150136017399099233716, −8.699652355872805014378056311266, −8.018157573330729115437611906063, −6.97679568932008852716898869411, −6.53304043649234037802438120150, −5.33501497548049589343922054884, −5.16676465047181917368548300874, −3.54087847528499532628363336008, −2.20947869122043575601815065610, −1.18688640597848024160435624178,
0.46691027141751224710254300592, 2.16098837889267984349405378872, 3.80383437914955922038339995140, 4.45254317053463480037779491853, 5.07311613505613968098414054656, 6.07186286862135935116198617027, 6.97341559362176077310617020292, 7.72399742906478940231366344214, 8.856719060214097726453631153105, 9.829633339722218958053180204852