| L(s) = 1 | + 3-s − 5-s + 2.24·7-s + 9-s + 1.14·11-s − 2.24·13-s − 15-s + 2.67·17-s + 5.34·19-s + 2.24·21-s − 2.67·23-s + 25-s + 27-s + 0.715·29-s + 31-s + 1.14·33-s − 2.24·35-s + 6.53·37-s − 2.24·39-s − 3.14·41-s − 12.2·43-s − 45-s + 10.8·47-s − 1.96·49-s + 2.67·51-s + 11.5·53-s − 1.14·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.848·7-s + 0.333·9-s + 0.344·11-s − 0.622·13-s − 0.258·15-s + 0.648·17-s + 1.22·19-s + 0.489·21-s − 0.557·23-s + 0.200·25-s + 0.192·27-s + 0.132·29-s + 0.179·31-s + 0.199·33-s − 0.379·35-s + 1.07·37-s − 0.359·39-s − 0.491·41-s − 1.86·43-s − 0.149·45-s + 1.58·47-s − 0.280·49-s + 0.374·51-s + 1.58·53-s − 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.811871430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.811871430\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2.24T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 - 0.715T + 29T^{2} \) |
| 37 | \( 1 - 6.53T + 37T^{2} \) |
| 41 | \( 1 + 3.14T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 2.71T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 - 3.75T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 0.384T + 83T^{2} \) |
| 89 | \( 1 + 2.62T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.80558038775716669567579980589, −7.45497876752233407989397025783, −6.69333408640981534234942318573, −5.66736023395041101560531102209, −5.01410942417554372059712424592, −4.28252549157512214805938041308, −3.53929560517869347805469075174, −2.74860287181644136617916102452, −1.80123511186182102379915962571, −0.854233977983507907235061486468,
0.854233977983507907235061486468, 1.80123511186182102379915962571, 2.74860287181644136617916102452, 3.53929560517869347805469075174, 4.28252549157512214805938041308, 5.01410942417554372059712424592, 5.66736023395041101560531102209, 6.69333408640981534234942318573, 7.45497876752233407989397025783, 7.80558038775716669567579980589
