| L(s) = 1 | + 3·3-s + 7-s + 6·9-s + 2·11-s + 13-s + 3·17-s − 19-s + 3·21-s − 3·23-s − 5·25-s + 9·27-s − 3·29-s − 8·31-s + 6·33-s + 10·37-s + 3·39-s − 12·41-s + 8·43-s + 8·47-s − 6·49-s + 9·51-s + 9·53-s − 3·57-s − 5·59-s − 10·61-s + 6·63-s + 7·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s + 0.603·11-s + 0.277·13-s + 0.727·17-s − 0.229·19-s + 0.654·21-s − 0.625·23-s − 25-s + 1.73·27-s − 0.557·29-s − 1.43·31-s + 1.04·33-s + 1.64·37-s + 0.480·39-s − 1.87·41-s + 1.21·43-s + 1.16·47-s − 6/7·49-s + 1.26·51-s + 1.23·53-s − 0.397·57-s − 0.650·59-s − 1.28·61-s + 0.755·63-s + 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.259109801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.259109801\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.415861681681162579943387548290, −9.027914182722105856770690690230, −7.943968172182858592403142779428, −7.74236222044461134007095453218, −6.60845811420238685670203067170, −5.46776906443032883183491997236, −4.08931083553499496867238827018, −3.63803508541238097399374767207, −2.43573806798025905726670590752, −1.52291178817025335850558392010,
1.52291178817025335850558392010, 2.43573806798025905726670590752, 3.63803508541238097399374767207, 4.08931083553499496867238827018, 5.46776906443032883183491997236, 6.60845811420238685670203067170, 7.74236222044461134007095453218, 7.943968172182858592403142779428, 9.027914182722105856770690690230, 9.415861681681162579943387548290
