| L(s) = 1 | + 1.41·5-s − 1.41·13-s + 2·17-s − 2.99·25-s − 4.24·29-s + 7.07·37-s − 10·41-s − 7·49-s − 12.7·53-s − 1.41·61-s − 2.00·65-s − 6·73-s + 2.82·85-s + 16·89-s − 8·97-s + 12.7·101-s − 9.89·109-s + 16·113-s + ⋯ |
| L(s) = 1 | + 0.632·5-s − 0.392·13-s + 0.485·17-s − 0.599·25-s − 0.787·29-s + 1.16·37-s − 1.56·41-s − 49-s − 1.74·53-s − 0.181·61-s − 0.248·65-s − 0.702·73-s + 0.306·85-s + 1.69·89-s − 0.812·97-s + 1.26·101-s − 0.948·109-s + 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.52109404702928365301644846476, −6.54087314687781178646132869587, −6.08717903503791125033654678799, −5.29588564817080454452227898866, −4.72925479392095584390529685450, −3.77513555973402075244308886001, −3.02925777274925323984776681687, −2.11966916343854302979676967310, −1.37344027613478459700621173591, 0,
1.37344027613478459700621173591, 2.11966916343854302979676967310, 3.02925777274925323984776681687, 3.77513555973402075244308886001, 4.72925479392095584390529685450, 5.29588564817080454452227898866, 6.08717903503791125033654678799, 6.54087314687781178646132869587, 7.52109404702928365301644846476
