| L(s) = 1 | − 5-s − 3·7-s + 11-s + 13-s + 3·17-s + 2·19-s + 5·23-s + 25-s + 6·29-s − 10·31-s + 3·35-s + 5·37-s − 3·41-s − 4·43-s + 6·47-s + 2·49-s − 5·53-s − 55-s − 8·59-s + 61-s − 65-s − 12·67-s + 71-s − 10·73-s − 3·77-s + 79-s − 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 0.301·11-s + 0.277·13-s + 0.727·17-s + 0.458·19-s + 1.04·23-s + 1/5·25-s + 1.11·29-s − 1.79·31-s + 0.507·35-s + 0.821·37-s − 0.468·41-s − 0.609·43-s + 0.875·47-s + 2/7·49-s − 0.686·53-s − 0.134·55-s − 1.04·59-s + 0.128·61-s − 0.124·65-s − 1.46·67-s + 0.118·71-s − 1.17·73-s − 0.341·77-s + 0.112·79-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.480626540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.480626540\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−7.55182154709733731282925872184, −7.09112523872932676561926379058, −6.35499315805336524102715849554, −5.75930364613115086570348204049, −4.93411850643084599214978203919, −4.12183045523383470544313157238, −3.25034829781988260087050532945, −2.99789940770590095024287331470, −1.61397926603679017391495385614, −0.59619038336214709161452591849,
0.59619038336214709161452591849, 1.61397926603679017391495385614, 2.99789940770590095024287331470, 3.25034829781988260087050532945, 4.12183045523383470544313157238, 4.93411850643084599214978203919, 5.75930364613115086570348204049, 6.35499315805336524102715849554, 7.09112523872932676561926379058, 7.55182154709733731282925872184
