| L(s) = 1 | + 4·11-s + 2·13-s − 6·17-s − 4·19-s + 23-s + 2·29-s + 2·37-s − 10·41-s − 4·43-s − 7·49-s + 6·53-s − 4·59-s − 10·61-s − 12·67-s − 8·71-s − 10·73-s + 8·79-s + 4·83-s − 18·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.208·23-s + 0.371·29-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 49-s + 0.824·53-s − 0.520·59-s − 1.28·61-s − 1.46·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s + 0.439·83-s − 1.90·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.571879264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.571879264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.82324917889207, −13.43244448321799, −13.14374608675127, −12.42292467301843, −11.88173500610804, −11.58186046667871, −10.87821831073782, −10.64777883551879, −9.937271472938617, −9.361387342229931, −8.799217848267564, −8.607074595978768, −7.975042790067582, −7.172790008847788, −6.680950678462847, −6.370240169260920, −5.829984232632385, −5.003921754574432, −4.350917671088369, −4.142557085999881, −3.287210112000598, −2.765816156836010, −1.756836036669573, −1.531046776178817, −0.3953555816073852,
0.3953555816073852, 1.531046776178817, 1.756836036669573, 2.765816156836010, 3.287210112000598, 4.142557085999881, 4.350917671088369, 5.003921754574432, 5.829984232632385, 6.370240169260920, 6.680950678462847, 7.172790008847788, 7.975042790067582, 8.607074595978768, 8.799217848267564, 9.361387342229931, 9.937271472938617, 10.64777883551879, 10.87821831073782, 11.58186046667871, 11.88173500610804, 12.42292467301843, 13.14374608675127, 13.43244448321799, 13.82324917889207
