L(s) = 1 | − 3-s − 2·5-s + 2·7-s + 9-s + 4·13-s + 2·15-s − 17-s + 2·19-s − 2·21-s + 2·23-s − 25-s − 27-s + 2·29-s − 4·31-s − 4·35-s − 6·37-s − 4·39-s − 6·41-s + 2·43-s − 2·45-s − 3·49-s + 51-s + 12·53-s − 2·57-s + 14·59-s + 6·61-s + 2·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.10·13-s + 0.516·15-s − 0.242·17-s + 0.458·19-s − 0.436·21-s + 0.417·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.676·35-s − 0.986·37-s − 0.640·39-s − 0.937·41-s + 0.304·43-s − 0.298·45-s − 3/7·49-s + 0.140·51-s + 1.64·53-s − 0.264·57-s + 1.82·59-s + 0.768·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.216916691\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.216916691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.24021368925312, −11.94751815371842, −11.44741666325173, −11.21016018797538, −10.83636874775663, −10.28745300874476, −9.857885026595103, −9.232169411533594, −8.624515870472450, −8.359470207187969, −8.007456650067527, −7.282168650933514, −6.977534281232258, −6.615065898366862, −5.749619428041195, −5.538491095914108, −5.001410805502971, −4.447295893004563, −3.930045926565628, −3.586342444894644, −3.018444672995713, −2.111236615437680, −1.687876267238762, −0.9201547119533036, −0.4901885569692777,
0.4901885569692777, 0.9201547119533036, 1.687876267238762, 2.111236615437680, 3.018444672995713, 3.586342444894644, 3.930045926565628, 4.447295893004563, 5.001410805502971, 5.538491095914108, 5.749619428041195, 6.615065898366862, 6.977534281232258, 7.282168650933514, 8.007456650067527, 8.359470207187969, 8.624515870472450, 9.232169411533594, 9.857885026595103, 10.28745300874476, 10.83636874775663, 11.21016018797538, 11.44741666325173, 11.94751815371842, 12.24021368925312