L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 1.41i·5-s + 2.82i·8-s − 2.00·10-s + 4·13-s + 4.00·16-s + 7.07i·17-s + 2.82i·20-s + 2.99·25-s − 5.65i·26-s + 9.89i·29-s − 5.65i·32-s + 10.0·34-s + 2·37-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s − 0.632i·5-s + 1.00i·8-s − 0.632·10-s + 1.10·13-s + 1.00·16-s + 1.71i·17-s + 0.632i·20-s + 0.599·25-s − 1.10i·26-s + 1.83i·29-s − 1.00i·32-s + 1.71·34-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.561206045\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.561206045\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 18.3iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.083559597761376916998406488775, −8.604615948543984806954555397492, −8.031822174114039517254103200728, −6.69117887297242671851382966503, −5.72250297994792867685532902065, −4.93933073014641451114900035031, −3.96120476397456095112552159462, −3.31426668407820043049485753548, −1.90683177118357553204744260386, −1.03503149258991882045679211844,
0.78090895439012213134471147038, 2.63994859600334458191895790052, 3.67887499882034198321008981467, 4.60221345991105671705217530867, 5.53493084200743304160227091114, 6.34394468186259511321388436319, 6.96859277718540326727871250078, 7.75092787742152422470158020981, 8.477249383934390045871409295796, 9.353836263512789625928165935555