| L(s) = 1 | − 3-s − 5-s − 2·9-s + 2·11-s + 5·13-s + 15-s − 4·17-s − 2·19-s + 23-s + 25-s + 5·27-s + 3·29-s − 7·31-s − 2·33-s + 2·37-s − 5·39-s − 9·41-s − 4·43-s + 2·45-s + 9·47-s − 7·49-s + 4·51-s + 6·53-s − 2·55-s + 2·57-s − 2·61-s − 5·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.603·11-s + 1.38·13-s + 0.258·15-s − 0.970·17-s − 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 1.25·31-s − 0.348·33-s + 0.328·37-s − 0.800·39-s − 1.40·41-s − 0.609·43-s + 0.298·45-s + 1.31·47-s − 49-s + 0.560·51-s + 0.824·53-s − 0.269·55-s + 0.264·57-s − 0.256·61-s − 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.51282980127230844041269754162, −6.57259955618011446432354032044, −6.35767815900295924967002307264, −5.48141761704551174444122937443, −4.75448749320085702861476856611, −3.90202612496681922874080243166, −3.32798042816200535822847402720, −2.20575522053657701332358872144, −1.12145775198436889524504268522, 0,
1.12145775198436889524504268522, 2.20575522053657701332358872144, 3.32798042816200535822847402720, 3.90202612496681922874080243166, 4.75448749320085702861476856611, 5.48141761704551174444122937443, 6.35767815900295924967002307264, 6.57259955618011446432354032044, 7.51282980127230844041269754162
