L(s) = 1 | − 2·4-s − 5·13-s + 4·16-s − 8·19-s − 5·25-s − 11·31-s − 10·37-s + 5·43-s + 10·52-s + 61-s − 8·64-s + 11·67-s − 17·73-s + 16·76-s − 13·79-s − 14·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 4-s − 1.38·13-s + 16-s − 1.83·19-s − 25-s − 1.97·31-s − 1.64·37-s + 0.762·43-s + 1.38·52-s + 0.128·61-s − 64-s + 1.34·67-s − 1.98·73-s + 1.83·76-s − 1.46·79-s − 1.42·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.76610904490892, −12.42674079997781, −11.95870069967214, −11.33556003119963, −10.88259555400170, −10.29649760616796, −10.06020777079136, −9.552327153716495, −9.047116671692975, −8.654568550406315, −8.346261521527448, −7.547679992575562, −7.389305702965647, −6.806110668375998, −6.135228136007735, −5.625256253719429, −5.269750542601105, −4.686152104060316, −4.239623323815198, −3.830084432021051, −3.287290747213561, −2.526886786396644, −1.988846603071698, −1.519911990354193, −0.4175045727371186, 0,
0.4175045727371186, 1.519911990354193, 1.988846603071698, 2.526886786396644, 3.287290747213561, 3.830084432021051, 4.239623323815198, 4.686152104060316, 5.269750542601105, 5.625256253719429, 6.135228136007735, 6.806110668375998, 7.389305702965647, 7.547679992575562, 8.346261521527448, 8.654568550406315, 9.047116671692975, 9.552327153716495, 10.06020777079136, 10.29649760616796, 10.88259555400170, 11.33556003119963, 11.95870069967214, 12.42674079997781, 12.76610904490892