| L(s) = 1 | − 3-s − 4·7-s + 9-s + 13-s − 2·17-s − 8·19-s + 4·21-s − 8·23-s − 27-s + 2·29-s + 4·31-s − 10·37-s − 39-s + 2·41-s − 4·43-s + 12·47-s + 9·49-s + 2·51-s + 6·53-s + 8·57-s + 2·61-s − 4·63-s + 8·67-s + 8·69-s − 12·71-s − 10·73-s − 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.485·17-s − 1.83·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 1.64·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s + 1.05·57-s + 0.256·61-s − 0.503·63-s + 0.977·67-s + 0.963·69-s − 1.42·71-s − 1.17·73-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.86457897823593, −14.15277193812616, −13.64048629410057, −13.23010837887524, −12.69526536888848, −12.26763666089567, −11.91288484582904, −11.21035459879078, −10.54122528526640, −10.21344341681205, −9.926451265491385, −9.075117548951101, −8.667011179541290, −8.193511630072473, −7.287523731487965, −6.815440258382812, −6.369150953571118, −5.948266613126758, −5.454213094267901, −4.469377647802336, −4.083599435780209, −3.569667861626527, −2.662606767037531, −2.183597634812718, −1.243373390138706, 0, 0,
1.243373390138706, 2.183597634812718, 2.662606767037531, 3.569667861626527, 4.083599435780209, 4.469377647802336, 5.454213094267901, 5.948266613126758, 6.369150953571118, 6.815440258382812, 7.287523731487965, 8.193511630072473, 8.667011179541290, 9.075117548951101, 9.926451265491385, 10.21344341681205, 10.54122528526640, 11.21035459879078, 11.91288484582904, 12.26763666089567, 12.69526536888848, 13.23010837887524, 13.64048629410057, 14.15277193812616, 14.86457897823593
