| L(s) = 1 | + 3-s + 3.80·5-s + 7-s + 9-s − 11-s + 3.80·13-s + 3.80·15-s + 0.334·17-s − 8.13·19-s + 21-s + 1.66·23-s + 9.47·25-s + 27-s + 0.195·29-s + 9.94·31-s − 33-s + 3.80·35-s − 4.47·37-s + 3.80·39-s − 6.27·41-s − 2.33·43-s + 3.80·45-s + 12.1·47-s + 49-s + 0.334·51-s + 7.94·53-s − 3.80·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.70·5-s + 0.377·7-s + 0.333·9-s − 0.301·11-s + 1.05·13-s + 0.982·15-s + 0.0812·17-s − 1.86·19-s + 0.218·21-s + 0.347·23-s + 1.89·25-s + 0.192·27-s + 0.0363·29-s + 1.78·31-s − 0.174·33-s + 0.643·35-s − 0.735·37-s + 0.609·39-s − 0.980·41-s − 0.356·43-s + 0.567·45-s + 1.77·47-s + 0.142·49-s + 0.0468·51-s + 1.09·53-s − 0.513·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.630934665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.630934665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3.80T + 5T^{2} \) |
| 13 | \( 1 - 3.80T + 13T^{2} \) |
| 17 | \( 1 - 0.334T + 17T^{2} \) |
| 19 | \( 1 + 8.13T + 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 29 | \( 1 - 0.195T + 29T^{2} \) |
| 31 | \( 1 - 9.94T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 6.27T + 41T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 7.94T + 53T^{2} \) |
| 59 | \( 1 + 3.74T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 0.139T + 67T^{2} \) |
| 71 | \( 1 + 4.66T + 71T^{2} \) |
| 73 | \( 1 - 4.19T + 73T^{2} \) |
| 79 | \( 1 + 3.33T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 9.88T + 89T^{2} \) |
| 97 | \( 1 - 0.0560T + 97T^{2} \) |
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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
−8.735057711871713540142184902534, −8.017936250755405139104662029455, −6.83077806371602115281747400800, −6.34090413913020710440486006447, −5.60379800454200075744989025950, −4.78867191690572264406367641038, −3.87433937635137962945145435742, −2.70407708168988341147495353155, −2.08602165387915773063808241065, −1.18987446753863425360299718142,
1.18987446753863425360299718142, 2.08602165387915773063808241065, 2.70407708168988341147495353155, 3.87433937635137962945145435742, 4.78867191690572264406367641038, 5.60379800454200075744989025950, 6.34090413913020710440486006447, 6.83077806371602115281747400800, 8.017936250755405139104662029455, 8.735057711871713540142184902534
