| L(s) = 1 | − 2-s − 3.39·3-s + 4-s + 1.47·5-s + 3.39·6-s − 7-s − 8-s + 8.52·9-s − 1.47·10-s − 2.86·11-s − 3.39·12-s − 3.17·13-s + 14-s − 5.00·15-s + 16-s − 1.91·17-s − 8.52·18-s − 1.17·19-s + 1.47·20-s + 3.39·21-s + 2.86·22-s + 3.17·23-s + 3.39·24-s − 2.82·25-s + 3.17·26-s − 18.7·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.95·3-s + 0.5·4-s + 0.659·5-s + 1.38·6-s − 0.377·7-s − 0.353·8-s + 2.84·9-s − 0.466·10-s − 0.864·11-s − 0.979·12-s − 0.881·13-s + 0.267·14-s − 1.29·15-s + 0.250·16-s − 0.463·17-s − 2.00·18-s − 0.270·19-s + 0.329·20-s + 0.740·21-s + 0.611·22-s + 0.662·23-s + 0.692·24-s − 0.565·25-s + 0.623·26-s − 3.60·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4806919921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4806919921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 3.39T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 8.13T + 29T^{2} \) |
| 31 | \( 1 - 9.00T + 31T^{2} \) |
| 37 | \( 1 - 4.78T + 37T^{2} \) |
| 43 | \( 1 + 1.78T + 43T^{2} \) |
| 47 | \( 1 - 4.66T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 4.69T + 61T^{2} \) |
| 67 | \( 1 + 1.38T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 0.478T + 79T^{2} \) |
| 83 | \( 1 + 2.30T + 83T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 + 9.95T + 97T^{2} \) |
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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
−10.45080691186608308882430168465, −10.21856686016366830837000525663, −9.357471152594815892820582700031, −7.893023572216397360725236312558, −6.86554318761368352250794253374, −6.27329991151756514975737190268, −5.37130606536787360280416168736, −4.52502161207761104859134602259, −2.39820279324503093475235230142, −0.73367515690551728798983658049,
0.73367515690551728798983658049, 2.39820279324503093475235230142, 4.52502161207761104859134602259, 5.37130606536787360280416168736, 6.27329991151756514975737190268, 6.86554318761368352250794253374, 7.893023572216397360725236312558, 9.357471152594815892820582700031, 10.21856686016366830837000525663, 10.45080691186608308882430168465
