| L(s) = 1 | − 2i·5-s − 4·7-s − 2i·11-s − 2i·13-s + 2·17-s + 2i·19-s − 4·23-s + 25-s − 6i·29-s + 8i·35-s + 10i·37-s − 6·41-s − 6i·43-s − 8·47-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s − 1.51·7-s − 0.603i·11-s − 0.554i·13-s + 0.485·17-s + 0.458i·19-s − 0.834·23-s + 0.200·25-s − 1.11i·29-s + 1.35i·35-s + 1.64i·37-s − 0.937·41-s − 0.914i·43-s − 1.16·47-s + 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 14iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.537883598187149394139660939031, −7.927597082960768217010357813251, −6.88483364047604802017468384205, −6.05413092799802394714867394531, −5.54545642580210773785881917936, −4.43840339046971541107811169446, −3.51512988593556574794619540593, −2.76949698107409344784867387252, −1.21146561810893222769175172833, 0,
1.89489278127510207413757070938, 3.02134894909850940868110394782, 3.54504386175201089083978945511, 4.65041292432824869473491905227, 5.75245531885267310765400026046, 6.63217600020591710133260327071, 6.90658693472445454577262700351, 7.78807401181537754550940713269, 8.901283822911148933219274961961
