| L(s) = 1 | + 3·3-s + 5·5-s − 7·7-s + 9·9-s + 22·11-s − 44·13-s + 15·15-s − 110·17-s − 22·19-s − 21·21-s − 36·23-s + 25·25-s + 27·27-s − 122·29-s − 186·31-s + 66·33-s − 35·35-s + 306·37-s − 132·39-s − 330·41-s + 20·43-s + 45·45-s − 64·47-s + 49·49-s − 330·51-s + 504·53-s + 110·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.938·13-s + 0.258·15-s − 1.56·17-s − 0.265·19-s − 0.218·21-s − 0.326·23-s + 1/5·25-s + 0.192·27-s − 0.781·29-s − 1.07·31-s + 0.348·33-s − 0.169·35-s + 1.35·37-s − 0.541·39-s − 1.25·41-s + 0.0709·43-s + 0.149·45-s − 0.198·47-s + 1/7·49-s − 0.906·51-s + 1.30·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 44 T + p^{3} T^{2} \) |
| 17 | \( 1 + 110 T + p^{3} T^{2} \) |
| 19 | \( 1 + 22 T + p^{3} T^{2} \) |
| 23 | \( 1 + 36 T + p^{3} T^{2} \) |
| 29 | \( 1 + 122 T + p^{3} T^{2} \) |
| 31 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 37 | \( 1 - 306 T + p^{3} T^{2} \) |
| 41 | \( 1 + 330 T + p^{3} T^{2} \) |
| 43 | \( 1 - 20 T + p^{3} T^{2} \) |
| 47 | \( 1 + 64 T + p^{3} T^{2} \) |
| 53 | \( 1 - 504 T + p^{3} T^{2} \) |
| 59 | \( 1 + 560 T + p^{3} T^{2} \) |
| 61 | \( 1 + 418 T + p^{3} T^{2} \) |
| 67 | \( 1 + 452 T + p^{3} T^{2} \) |
| 71 | \( 1 + 146 T + p^{3} T^{2} \) |
| 73 | \( 1 + 236 T + p^{3} T^{2} \) |
| 79 | \( 1 - 536 T + p^{3} T^{2} \) |
| 83 | \( 1 + 92 T + p^{3} T^{2} \) |
| 89 | \( 1 + 574 T + p^{3} T^{2} \) |
| 97 | \( 1 - 184 T + p^{3} T^{2} \) |
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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
−9.280006060389995086821092815219, −8.805565892310133215201807635645, −7.62471405727836087989529968364, −6.83827543716163066507640561175, −6.00472364433681541136988791645, −4.77922430864433375372275550165, −3.87062882138216773831928685756, −2.64735697558271889589125359580, −1.75869645659215497190647513550, 0,
1.75869645659215497190647513550, 2.64735697558271889589125359580, 3.87062882138216773831928685756, 4.77922430864433375372275550165, 6.00472364433681541136988791645, 6.83827543716163066507640561175, 7.62471405727836087989529968364, 8.805565892310133215201807635645, 9.280006060389995086821092815219
