| L(s) = 1 | + 3-s + 5-s + 2·7-s − 2·9-s − 13-s + 15-s − 4·17-s − 4·19-s + 2·21-s − 23-s + 25-s − 5·27-s + 3·29-s + 31-s + 2·35-s + 8·37-s − 39-s − 5·41-s − 6·43-s − 2·45-s − 9·47-s − 3·49-s − 4·51-s − 2·53-s − 4·57-s − 4·63-s − 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s − 0.277·13-s + 0.258·15-s − 0.970·17-s − 0.917·19-s + 0.436·21-s − 0.208·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 0.179·31-s + 0.338·35-s + 1.31·37-s − 0.160·39-s − 0.780·41-s − 0.914·43-s − 0.298·45-s − 1.31·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s − 0.529·57-s − 0.503·63-s − 0.124·65-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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.82265513814135020546080988419, −6.72360955383732768713468315340, −6.31258134437019562477247879479, −5.35129885972936414903551070680, −4.73457956302674779066356414490, −3.97994801842954418872803416043, −2.94096251000949591919791059585, −2.30026497336620187954159958479, −1.53609766383920065995118888181, 0,
1.53609766383920065995118888181, 2.30026497336620187954159958479, 2.94096251000949591919791059585, 3.97994801842954418872803416043, 4.73457956302674779066356414490, 5.35129885972936414903551070680, 6.31258134437019562477247879479, 6.72360955383732768713468315340, 7.82265513814135020546080988419
