| L(s) = 1 | + 3-s − 2·4-s − 3.46·5-s + 9-s + 2.26·11-s − 2·12-s + 1.46·13-s − 3.46·15-s + 4·16-s − 5·17-s − 7.46·19-s + 6.92·20-s − 1.46·23-s + 6.99·25-s + 27-s + 5.19·29-s − 2.26·31-s + 2.26·33-s − 2·36-s − 37-s + 1.46·39-s + 41-s − 3.92·43-s − 4.53·44-s − 3.46·45-s − 1.92·47-s + 4·48-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 1.54·5-s + 0.333·9-s + 0.683·11-s − 0.577·12-s + 0.406·13-s − 0.894·15-s + 16-s − 1.21·17-s − 1.71·19-s + 1.54·20-s − 0.305·23-s + 1.39·25-s + 0.192·27-s + 0.964·29-s − 0.407·31-s + 0.394·33-s − 0.333·36-s − 0.164·37-s + 0.234·39-s + 0.156·41-s − 0.599·43-s − 0.683·44-s − 0.516·45-s − 0.281·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8320492335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8320492335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 - 5.19T + 29T^{2} \) |
| 31 | \( 1 + 2.26T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 43 | \( 1 + 3.92T + 43T^{2} \) |
| 47 | \( 1 + 1.92T + 47T^{2} \) |
| 53 | \( 1 + 2.92T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 - 0.535T + 67T^{2} \) |
| 71 | \( 1 - 2.80T + 71T^{2} \) |
| 73 | \( 1 + 1.19T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.332084222414555977303129528652, −7.55720524177333256665773263579, −6.76046695167375716021199324542, −6.07985993550982072504098221293, −4.70148630756669300876138364389, −4.37617718652479557286822383532, −3.80748329693678630939012360002, −3.08415766596524700145753616135, −1.80093794891218194311150207312, −0.46538309134254912408897046436,
0.46538309134254912408897046436, 1.80093794891218194311150207312, 3.08415766596524700145753616135, 3.80748329693678630939012360002, 4.37617718652479557286822383532, 4.70148630756669300876138364389, 6.07985993550982072504098221293, 6.76046695167375716021199324542, 7.55720524177333256665773263579, 8.332084222414555977303129528652
