| L(s) = 1 | − 3-s + (−1 + 2i)5-s + 2i·7-s + 9-s − 6i·11-s − 2·13-s + (1 − 2i)15-s − 6i·17-s + 4i·19-s − 2i·21-s + 8i·23-s + (−3 − 4i)25-s − 27-s + 8·31-s + 6i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.447 + 0.894i)5-s + 0.755i·7-s + 0.333·9-s − 1.80i·11-s − 0.554·13-s + (0.258 − 0.516i)15-s − 1.45i·17-s + 0.917i·19-s − 0.436i·21-s + 1.66i·23-s + (−0.600 − 0.800i)25-s − 0.192·27-s + 1.43·31-s + 1.04i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9079178714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9079178714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.670847643029248028092444877962, −7.80427070371199971056721954318, −7.34038455312959905105964029282, −6.28905735943566354378836079933, −5.85994088413964385767583043704, −5.14936883304616687481552464239, −4.04949561136821546815550243144, −3.12822481043466951292579718225, −2.56799039311893495783023479249, −0.967272630754393992379054460501,
0.35898832583721480186938109754, 1.47394261365309494293767836864, 2.53957795830406123174291566226, 4.11931217473419772273132318954, 4.43301736116894131327615038362, 4.97857079798671250960610203551, 6.11428579330236840274088977616, 6.85949254509761650764275651052, 7.49413924637445370228607127530, 8.165804499750410969413228790240
