| L(s) = 1 | + 2i·2-s − 4·4-s + 4i·7-s − 8i·8-s − 12·11-s − 58i·13-s − 8·14-s + 16·16-s + 66i·17-s + 100·19-s − 24i·22-s − 132i·23-s + 116·26-s − 16i·28-s − 90·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.215i·7-s − 0.353i·8-s − 0.328·11-s − 1.23i·13-s − 0.152·14-s + 0.250·16-s + 0.941i·17-s + 1.20·19-s − 0.232i·22-s − 1.19i·23-s + 0.874·26-s − 0.107i·28-s − 0.576·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.718524662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718524662\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4iT - 343T^{2} \) |
| 11 | \( 1 + 12T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 100T + 6.85e3T^{2} \) |
| 23 | \( 1 + 132iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 152T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 438T + 6.89e4T^{2} \) |
| 43 | \( 1 - 32iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 204iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 222iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 420T + 2.05e5T^{2} \) |
| 61 | \( 1 - 902T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.02e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 432T + 3.57e5T^{2} \) |
| 73 | \( 1 - 362iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 160T + 4.93e5T^{2} \) |
| 83 | \( 1 + 72iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 810T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3iT - 9.12e5T^{2} \) |
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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
−10.51804494624323660398869690701, −9.867074630777899289913721191661, −8.671545843867980458318146727698, −8.019383443235623041770190810278, −7.09416946760370309077869307043, −5.93781691166390590797326251258, −5.26132311540951957578189144581, −3.98117166468712184266240546502, −2.67582355966347299025569646735, −0.75602300209059065478118762817,
0.959846312424796667948307735738, 2.34921737102649215019836779592, 3.56924481079303792323852415316, 4.65756054180351006520017865160, 5.68093892312792666205441509627, 7.05372382141228285960933022142, 7.85307847814056745705920043101, 9.241127417908519745188081306337, 9.560657420168125583345953404815, 10.72848523748022034471888247187
