L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 3·11-s − 4·13-s − 15-s + 6·17-s − 19-s + 21-s + 23-s + 25-s + 27-s − 6·29-s − 4·31-s − 3·33-s − 35-s + 2·37-s − 4·39-s + 3·41-s + 2·43-s − 45-s + 9·47-s + 49-s + 6·51-s + 3·53-s + 3·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s − 0.258·15-s + 1.45·17-s − 0.229·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.522·33-s − 0.169·35-s + 0.328·37-s − 0.640·39-s + 0.468·41-s + 0.304·43-s − 0.149·45-s + 1.31·47-s + 1/7·49-s + 0.840·51-s + 0.412·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.54348190104469009841352579605, −6.99884354683835100586748326697, −5.73410264655960390494986239911, −5.35057865712725557347700646249, −4.48538765587776945557460765828, −3.80851839034297077713389006545, −2.93214329407990956416122281353, −2.34759014922454150931583714228, −1.28452771954649264874520991639, 0,
1.28452771954649264874520991639, 2.34759014922454150931583714228, 2.93214329407990956416122281353, 3.80851839034297077713389006545, 4.48538765587776945557460765828, 5.35057865712725557347700646249, 5.73410264655960390494986239911, 6.99884354683835100586748326697, 7.54348190104469009841352579605