L(s) = 1 | + 2·5-s − 3·9-s − 6·13-s − 2·17-s − 25-s − 10·29-s − 2·37-s − 10·41-s − 6·45-s + 14·53-s + 10·61-s − 12·65-s + 6·73-s + 9·81-s − 4·85-s − 10·89-s − 18·97-s + 2·101-s + 6·109-s − 14·113-s + 18·117-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 9-s − 1.66·13-s − 0.485·17-s − 1/5·25-s − 1.85·29-s − 0.328·37-s − 1.56·41-s − 0.894·45-s + 1.92·53-s + 1.28·61-s − 1.48·65-s + 0.702·73-s + 81-s − 0.433·85-s − 1.05·89-s − 1.82·97-s + 0.199·101-s + 0.574·109-s − 1.31·113-s + 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.157650363860643692962612117974, −8.325594584516082101612725346303, −7.36223546213665716124125288316, −6.61566651441991094272623466583, −5.53149385157427342467827793824, −5.20596475995087001705061173512, −3.88850761061225073398755969122, −2.65687059651135944196433512426, −1.95654116885815574508059039227, 0,
1.95654116885815574508059039227, 2.65687059651135944196433512426, 3.88850761061225073398755969122, 5.20596475995087001705061173512, 5.53149385157427342467827793824, 6.61566651441991094272623466583, 7.36223546213665716124125288316, 8.325594584516082101612725346303, 9.157650363860643692962612117974