| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 2·7-s + 3·8-s + 9-s + 10-s − 2·11-s − 12-s − 5·13-s + 2·14-s − 15-s − 16-s − 18-s − 8·19-s + 20-s − 2·21-s + 2·22-s + 23-s + 3·24-s − 4·25-s + 5·26-s + 27-s + 2·28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s − 1.38·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.436·21-s + 0.426·22-s + 0.208·23-s + 0.612·24-s − 4/5·25-s + 0.980·26-s + 0.192·27-s + 0.377·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−10.91310650137248910674296433054, −9.977343219212868982373122074796, −9.436339287145902394867629318382, −8.332830558753638476775725397728, −7.73611775760879899497175825486, −6.61986708268418450593402445280, −4.94768257430313977239388435598, −3.90959883042434151888908819851, −2.38565174811414294967341065866, 0,
2.38565174811414294967341065866, 3.90959883042434151888908819851, 4.94768257430313977239388435598, 6.61986708268418450593402445280, 7.73611775760879899497175825486, 8.332830558753638476775725397728, 9.436339287145902394867629318382, 9.977343219212868982373122074796, 10.91310650137248910674296433054
