| L(s) = 1 | + 3·3-s + 5-s + 6·9-s − 5·11-s − 5·13-s + 3·15-s + 7·17-s + 2·19-s + 2·23-s + 25-s + 9·27-s − 7·29-s + 4·31-s − 15·33-s + 6·37-s − 15·39-s + 12·41-s − 2·43-s + 6·45-s + 47-s + 21·51-s − 5·55-s + 6·57-s + 4·59-s + 4·61-s − 5·65-s + 8·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s − 1.50·11-s − 1.38·13-s + 0.774·15-s + 1.69·17-s + 0.458·19-s + 0.417·23-s + 1/5·25-s + 1.73·27-s − 1.29·29-s + 0.718·31-s − 2.61·33-s + 0.986·37-s − 2.40·39-s + 1.87·41-s − 0.304·43-s + 0.894·45-s + 0.145·47-s + 2.94·51-s − 0.674·55-s + 0.794·57-s + 0.520·59-s + 0.512·61-s − 0.620·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.620603158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.620603158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.92569025719452, −15.13373486288676, −14.87754304979238, −14.32019758386306, −13.93048416453304, −13.23083440908293, −12.80938037750294, −12.42855178637713, −11.53371487328102, −10.66633071577337, −10.05026621923427, −9.626105589332534, −9.360696339588295, −8.395950110455690, −7.892888934450544, −7.509245684830943, −7.084403983715135, −5.849658710998730, −5.307515950576888, −4.625848780164993, −3.750772724612488, −2.935424483826749, −2.630030531608772, −1.949236212584872, −0.8399407915507089,
0.8399407915507089, 1.949236212584872, 2.630030531608772, 2.935424483826749, 3.750772724612488, 4.625848780164993, 5.307515950576888, 5.849658710998730, 7.084403983715135, 7.509245684830943, 7.892888934450544, 8.395950110455690, 9.360696339588295, 9.626105589332534, 10.05026621923427, 10.66633071577337, 11.53371487328102, 12.42855178637713, 12.80938037750294, 13.23083440908293, 13.93048416453304, 14.32019758386306, 14.87754304979238, 15.13373486288676, 15.92569025719452
