| L(s) = 1 | + 7-s − 3·9-s − 11-s − 13-s + 17-s − 2·19-s − 5·25-s + 5·29-s + 31-s + 7·37-s − 9·41-s − 4·43-s − 8·47-s + 49-s − 6·53-s − 59-s + 9·61-s − 3·63-s − 5·67-s − 4·71-s + 11·73-s − 77-s + 4·79-s + 9·81-s − 6·83-s + 14·89-s − 91-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 0.301·11-s − 0.277·13-s + 0.242·17-s − 0.458·19-s − 25-s + 0.928·29-s + 0.179·31-s + 1.15·37-s − 1.40·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.130·59-s + 1.15·61-s − 0.377·63-s − 0.610·67-s − 0.474·71-s + 1.28·73-s − 0.113·77-s + 0.450·79-s + 81-s − 0.658·83-s + 1.48·89-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192556 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192556 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.163156137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163156137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.23040565208037, −12.60298403020590, −11.98492907753974, −11.75319209367719, −11.30769284010386, −10.74501983329690, −10.33851753967850, −9.717182906900375, −9.440552300889841, −8.687004197793597, −8.179646737937113, −8.087183234236493, −7.444666178856862, −6.657979527365733, −6.419340863469651, −5.712854531293466, −5.336990275012135, −4.715180394882594, −4.353401609955585, −3.427595837464583, −3.179908814565513, −2.365045247122814, −1.987962348361094, −1.149369767274542, −0.3160020098879390,
0.3160020098879390, 1.149369767274542, 1.987962348361094, 2.365045247122814, 3.179908814565513, 3.427595837464583, 4.353401609955585, 4.715180394882594, 5.336990275012135, 5.712854531293466, 6.419340863469651, 6.657979527365733, 7.444666178856862, 8.087183234236493, 8.179646737937113, 8.687004197793597, 9.440552300889841, 9.717182906900375, 10.33851753967850, 10.74501983329690, 11.30769284010386, 11.75319209367719, 11.98492907753974, 12.60298403020590, 13.23040565208037
