| L(s) = 1 | − 1.91·3-s − 5-s + 3.06·7-s + 0.659·9-s + 3.04·13-s + 1.91·15-s − 0.463·17-s + 7.89·19-s − 5.86·21-s + 1.39·23-s + 25-s + 4.47·27-s + 3.72·29-s + 10.4·31-s − 3.06·35-s + 1.84·37-s − 5.83·39-s − 4.40·41-s + 1.31·43-s − 0.659·45-s + 2.98·47-s + 2.40·49-s + 0.887·51-s + 4.18·53-s − 15.1·57-s − 2.81·59-s − 2.01·61-s + ⋯ |
| L(s) = 1 | − 1.10·3-s − 0.447·5-s + 1.15·7-s + 0.219·9-s + 0.845·13-s + 0.493·15-s − 0.112·17-s + 1.81·19-s − 1.28·21-s + 0.289·23-s + 0.200·25-s + 0.861·27-s + 0.691·29-s + 1.88·31-s − 0.518·35-s + 0.303·37-s − 0.934·39-s − 0.688·41-s + 0.200·43-s − 0.0983·45-s + 0.434·47-s + 0.343·49-s + 0.124·51-s + 0.575·53-s − 2.00·57-s − 0.366·59-s − 0.258·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.748094334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.748094334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.91T + 3T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 17 | \( 1 + 0.463T + 17T^{2} \) |
| 19 | \( 1 - 7.89T + 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 - 3.72T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 + 4.40T + 41T^{2} \) |
| 43 | \( 1 - 1.31T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 - 4.18T + 53T^{2} \) |
| 59 | \( 1 + 2.81T + 59T^{2} \) |
| 61 | \( 1 + 2.01T + 61T^{2} \) |
| 67 | \( 1 - 6.75T + 67T^{2} \) |
| 71 | \( 1 - 6.52T + 71T^{2} \) |
| 73 | \( 1 + 9.87T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 8.91T + 83T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−7.69343827205637051494037552139, −6.93011608514429340697016924628, −6.25263335107190140025421612202, −5.56475465572886219067609128676, −4.94241225488230705133492464330, −4.49120233507816627122435644212, −3.48995627903415740796845139834, −2.64943019945693075814562586483, −1.31132945221320451452275439074, −0.78423279656584149291302740495,
0.78423279656584149291302740495, 1.31132945221320451452275439074, 2.64943019945693075814562586483, 3.48995627903415740796845139834, 4.49120233507816627122435644212, 4.94241225488230705133492464330, 5.56475465572886219067609128676, 6.25263335107190140025421612202, 6.93011608514429340697016924628, 7.69343827205637051494037552139
