| L(s) = 1 | − 1.63·2-s − 0.492·3-s + 0.678·4-s − 1.90·5-s + 0.806·6-s + 0.625·7-s + 2.16·8-s − 2.75·9-s + 3.11·10-s + 1.86·11-s − 0.334·12-s − 3.61·13-s − 1.02·14-s + 0.937·15-s − 4.89·16-s − 0.946·17-s + 4.51·18-s − 0.390·19-s − 1.29·20-s − 0.308·21-s − 3.05·22-s + 23-s − 1.06·24-s − 1.38·25-s + 5.91·26-s + 2.83·27-s + 0.424·28-s + ⋯ |
| L(s) = 1 | − 1.15·2-s − 0.284·3-s + 0.339·4-s − 0.850·5-s + 0.329·6-s + 0.236·7-s + 0.764·8-s − 0.919·9-s + 0.984·10-s + 0.561·11-s − 0.0965·12-s − 1.00·13-s − 0.273·14-s + 0.241·15-s − 1.22·16-s − 0.229·17-s + 1.06·18-s − 0.0895·19-s − 0.288·20-s − 0.0672·21-s − 0.650·22-s + 0.208·23-s − 0.217·24-s − 0.276·25-s + 1.15·26-s + 0.545·27-s + 0.0802·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4502320836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4502320836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.63T + 2T^{2} \) |
| 3 | \( 1 + 0.492T + 3T^{2} \) |
| 5 | \( 1 + 1.90T + 5T^{2} \) |
| 7 | \( 1 - 0.625T + 7T^{2} \) |
| 11 | \( 1 - 1.86T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 + 0.946T + 17T^{2} \) |
| 19 | \( 1 + 0.390T + 19T^{2} \) |
| 31 | \( 1 - 5.69T + 31T^{2} \) |
| 37 | \( 1 - 5.54T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 0.753T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 - 3.41T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 - 7.92T + 67T^{2} \) |
| 71 | \( 1 + 4.28T + 71T^{2} \) |
| 73 | \( 1 + 7.27T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−10.45210835909598551519902891690, −9.508904429041783557001704223149, −8.810240174696533774422967672698, −7.964283418800231551994083254940, −7.39068108972250363125435101627, −6.25598451889362442181277938292, −4.95522406648140013575716903127, −4.03976017700842208517907818204, −2.45660221968565723091574690408, −0.67690630770963988258318948973,
0.67690630770963988258318948973, 2.45660221968565723091574690408, 4.03976017700842208517907818204, 4.95522406648140013575716903127, 6.25598451889362442181277938292, 7.39068108972250363125435101627, 7.964283418800231551994083254940, 8.810240174696533774422967672698, 9.508904429041783557001704223149, 10.45210835909598551519902891690
