| L(s) = 1 | + 2i·5-s + 4i·11-s + 2i·13-s − 2·17-s − 4i·19-s + 8·23-s + 25-s + 6i·29-s − 8·31-s + 6i·37-s − 6·41-s − 4i·43-s − 7·49-s + 2i·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 0.894i·5-s + 1.20i·11-s + 0.554i·13-s − 0.485·17-s − 0.917i·19-s + 1.66·23-s + 0.200·25-s + 1.11i·29-s − 1.43·31-s + 0.986i·37-s − 0.937·41-s − 0.609i·43-s − 49-s + 0.274i·53-s − 1.07·55-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
| \(L(1)\) |
\(\approx\) |
\(1.245064889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.245064889\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 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
−9.183613012755839829707821442260, −8.753317420517852031550437677128, −7.41254844927033011940088139374, −7.00442954320934135115804327993, −6.50700822368571191049958798507, −5.16519182346066886216240243061, −4.62101414134374887702865290048, −3.44394556910257389588949361227, −2.62659519019132278283014872433, −1.56549418357807959300628701511,
0.42676721229287966849756161539, 1.56115641518675559450081521212, 2.94754481817978253817324739577, 3.77996024055611700449390339397, 4.83395642531932385037076314908, 5.51965957782088520786944116422, 6.23240453467898262891057845754, 7.26805596659925573871003424356, 8.111317042625891461224272701422, 8.731291351606726848103343448412
