L(s) = 1 | − 2.75i·3-s + 3.84i·7-s + 1.41·9-s − 6.19i·11-s + 16.1·13-s − 5.20·17-s + 36.2i·19-s + 10.6·21-s − 22.0i·23-s − 28.6i·27-s + 20.0·29-s + 26.4i·31-s − 17.0·33-s + 69.3·37-s − 44.3i·39-s + ⋯ |
L(s) = 1 | − 0.918i·3-s + 0.549i·7-s + 0.157·9-s − 0.562i·11-s + 1.23·13-s − 0.306·17-s + 1.90i·19-s + 0.504·21-s − 0.958i·23-s − 1.06i·27-s + 0.690·29-s + 0.852i·31-s − 0.516·33-s + 1.87·37-s − 1.13i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.069642407\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.069642407\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.75iT - 9T^{2} \) |
| 7 | \( 1 - 3.84iT - 49T^{2} \) |
| 11 | \( 1 + 6.19iT - 121T^{2} \) |
| 13 | \( 1 - 16.1T + 169T^{2} \) |
| 17 | \( 1 + 5.20T + 289T^{2} \) |
| 19 | \( 1 - 36.2iT - 361T^{2} \) |
| 23 | \( 1 + 22.0iT - 529T^{2} \) |
| 29 | \( 1 - 20.0T + 841T^{2} \) |
| 31 | \( 1 - 26.4iT - 961T^{2} \) |
| 37 | \( 1 - 69.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 11.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 25.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 66.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 39.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 27.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 54.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 107. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 70.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 37.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 97.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 133.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 6.40T + 9.40e3T^{2} \) |
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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
−10.06617316948082995267546934919, −8.819558499625834289308724479410, −8.282063031483948974307607091846, −7.44242051297869911771879625975, −6.19420573947752110942148661984, −6.03922478618646135086809283908, −4.48714837638434181283359026593, −3.34255171861631631915988091512, −2.03413222821155893138646672702, −0.952716885536687175366369809747,
1.04665058182290120798737263806, 2.72951988459881387882698495541, 4.03952535439498710256315709127, 4.47824046182685693043400685853, 5.64128532399189099088698549714, 6.76804226275593960312021754749, 7.53315944194217920217168361584, 8.702510708851886544724216027752, 9.446910645135695683066234755207, 10.08520175886183248985261589869