| L(s) = 1 | + 2-s + 2.39·3-s + 4-s − 1.59·5-s + 2.39·6-s + 2.89·7-s + 8-s + 2.75·9-s − 1.59·10-s + 11-s + 2.39·12-s + 0.785·13-s + 2.89·14-s − 3.81·15-s + 16-s + 3.04·17-s + 2.75·18-s + 2.84·19-s − 1.59·20-s + 6.93·21-s + 22-s + 4.31·23-s + 2.39·24-s − 2.47·25-s + 0.785·26-s − 0.597·27-s + 2.89·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.38·3-s + 0.5·4-s − 0.711·5-s + 0.979·6-s + 1.09·7-s + 0.353·8-s + 0.916·9-s − 0.502·10-s + 0.301·11-s + 0.692·12-s + 0.217·13-s + 0.772·14-s − 0.984·15-s + 0.250·16-s + 0.738·17-s + 0.648·18-s + 0.653·19-s − 0.355·20-s + 1.51·21-s + 0.213·22-s + 0.900·23-s + 0.489·24-s − 0.494·25-s + 0.154·26-s − 0.114·27-s + 0.546·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.482329955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.482329955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 13 | \( 1 - 0.785T + 13T^{2} \) |
| 17 | \( 1 - 3.04T + 17T^{2} \) |
| 19 | \( 1 - 2.84T + 19T^{2} \) |
| 23 | \( 1 - 4.31T + 23T^{2} \) |
| 29 | \( 1 - 6.27T + 29T^{2} \) |
| 31 | \( 1 + 7.01T + 31T^{2} \) |
| 37 | \( 1 + 4.04T + 37T^{2} \) |
| 41 | \( 1 + 1.06T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 - 0.307T + 47T^{2} \) |
| 53 | \( 1 + 6.44T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 1.29T + 61T^{2} \) |
| 67 | \( 1 - 5.84T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 + 1.60T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 4.59T + 89T^{2} \) |
| 97 | \( 1 + 7.08T + 97T^{2} \) |
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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
−8.191687985246762231361310344522, −7.68532359315229154401330502312, −7.26150358830951946399235347069, −6.13944973168295982145488098088, −5.15522430269524702391298429649, −4.51475698976042471163802407811, −3.60188451349870800297806744073, −3.20692918050114893427639643041, −2.14685789616521069979533135995, −1.24842873151360844685614047895,
1.24842873151360844685614047895, 2.14685789616521069979533135995, 3.20692918050114893427639643041, 3.60188451349870800297806744073, 4.51475698976042471163802407811, 5.15522430269524702391298429649, 6.13944973168295982145488098088, 7.26150358830951946399235347069, 7.68532359315229154401330502312, 8.191687985246762231361310344522
