| L(s) = 1 | − 5-s − 2·7-s − 3·11-s − 13-s + 3·17-s − 8·19-s + 3·23-s + 25-s + 9·29-s + 7·31-s + 2·35-s + 2·37-s + 12·41-s + 7·43-s − 3·47-s − 3·49-s − 12·53-s + 3·55-s + 12·59-s − 10·61-s + 65-s + 4·67-s + 2·73-s + 6·77-s + 79-s + 18·83-s − 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.904·11-s − 0.277·13-s + 0.727·17-s − 1.83·19-s + 0.625·23-s + 1/5·25-s + 1.67·29-s + 1.25·31-s + 0.338·35-s + 0.328·37-s + 1.87·41-s + 1.06·43-s − 0.437·47-s − 3/7·49-s − 1.64·53-s + 0.404·55-s + 1.56·59-s − 1.28·61-s + 0.124·65-s + 0.488·67-s + 0.234·73-s + 0.683·77-s + 0.112·79-s + 1.97·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.180837031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.180837031\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 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
−9.054902793891064092582746405929, −8.179991731963249513899814243630, −7.68798259008681395333701987357, −6.60366619865162624235310481044, −6.12583377355996061020471857170, −4.94065208966695643772689925797, −4.27374684646969074066639414771, −3.12255554709506272654547929650, −2.43492436524524097349088307517, −0.68984650130164910647835476711,
0.68984650130164910647835476711, 2.43492436524524097349088307517, 3.12255554709506272654547929650, 4.27374684646969074066639414771, 4.94065208966695643772689925797, 6.12583377355996061020471857170, 6.60366619865162624235310481044, 7.68798259008681395333701987357, 8.179991731963249513899814243630, 9.054902793891064092582746405929
