| L(s) = 1 | + 3-s + 5-s + 3·7-s + 9-s + 2·11-s − 5·13-s + 15-s + 2·17-s + 5·19-s + 3·21-s + 8·23-s + 25-s + 27-s + 29-s − 8·31-s + 2·33-s + 3·35-s − 5·39-s − 8·41-s + 45-s − 6·47-s + 2·49-s + 2·51-s + 12·53-s + 2·55-s + 5·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.603·11-s − 1.38·13-s + 0.258·15-s + 0.485·17-s + 1.14·19-s + 0.654·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.185·29-s − 1.43·31-s + 0.348·33-s + 0.507·35-s − 0.800·39-s − 1.24·41-s + 0.149·45-s − 0.875·47-s + 2/7·49-s + 0.280·51-s + 1.64·53-s + 0.269·55-s + 0.662·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.164667568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.164667568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.53620755287894, −12.08736906826283, −11.83135910132682, −11.10631976250504, −10.96701966631971, −10.16040036321154, −9.819051442687944, −9.442634524885925, −8.919709745229433, −8.542468006688802, −8.013387965725754, −7.432841036811575, −7.093934772166532, −6.864859740014129, −5.938660553896741, −5.321707053689066, −5.116876738827760, −4.654857537932561, −4.002602343477725, −3.343766758638388, −2.923737481516108, −2.330240940097408, −1.635282692196460, −1.367765131206744, −0.5653457252816181,
0.5653457252816181, 1.367765131206744, 1.635282692196460, 2.330240940097408, 2.923737481516108, 3.343766758638388, 4.002602343477725, 4.654857537932561, 5.116876738827760, 5.321707053689066, 5.938660553896741, 6.864859740014129, 7.093934772166532, 7.432841036811575, 8.013387965725754, 8.542468006688802, 8.919709745229433, 9.442634524885925, 9.819051442687944, 10.16040036321154, 10.96701966631971, 11.10631976250504, 11.83135910132682, 12.08736906826283, 12.53620755287894
