L(s) = 1 | + (0.831 − 0.831i)2-s + (−0.581 + 1.40i)3-s − 0.381i·4-s + (0.684 + 1.65i)6-s + (−1.57 + 0.652i)7-s + (0.513 + 0.513i)8-s + (−0.928 − 0.928i)9-s + (0.536 + 0.222i)12-s + (−0.767 + 1.85i)14-s + 1.23·16-s + (−0.309 + 0.951i)17-s − 1.54·18-s − 2.59i·21-s + (−1.02 + 0.422i)24-s + (0.707 + 0.707i)25-s + ⋯ |
L(s) = 1 | + (0.831 − 0.831i)2-s + (−0.581 + 1.40i)3-s − 0.381i·4-s + (0.684 + 1.65i)6-s + (−1.57 + 0.652i)7-s + (0.513 + 0.513i)8-s + (−0.928 − 0.928i)9-s + (0.536 + 0.222i)12-s + (−0.767 + 1.85i)14-s + 1.23·16-s + (−0.309 + 0.951i)17-s − 1.54·18-s − 2.59i·21-s + (−1.02 + 0.422i)24-s + (0.707 + 0.707i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.020000971\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.020000971\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (0.309 - 0.951i)T \) |
| 47 | \( 1 + iT \) |
good | 2 | \( 1 + (-0.831 + 0.831i)T - iT^{2} \) |
| 3 | \( 1 + (0.581 - 1.40i)T + (-0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (1.57 - 0.652i)T + (0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.497 + 1.20i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-0.221 + 0.221i)T - iT^{2} \) |
| 59 | \( 1 + (-1.14 - 1.14i)T + iT^{2} \) |
| 61 | \( 1 + (-1.79 + 0.744i)T + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.763 + 1.84i)T + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.763 - 1.84i)T + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + (1 - i)T - iT^{2} \) |
| 89 | \( 1 + 1.17iT - T^{2} \) |
| 97 | \( 1 + (0.431 + 0.178i)T + (0.707 + 0.707i)T^{2} \) |
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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
−10.75556802865747950595435896802, −10.01966088943660178773291077865, −9.357777363898797653919917815141, −8.452694060189380969556852050973, −6.87091329499942075002468297117, −5.79183669810009959197487910181, −5.20123193370345540502669099253, −4.00886176312839353012338363869, −3.54773911631690299192247486021, −2.46756349797084567514259080872,
0.894786694864230422643487387847, 2.78373412992047753801095905578, 4.08830200386820838153504094096, 5.26714025412207548659826200789, 6.19942502306389733751434361438, 6.77534173904016297911675028650, 7.10318596731421927655803276581, 8.123786997797798011474079398576, 9.537028490037877585406052736239, 10.28334715799356289666740404040