| L(s) = 1 | + 0.0880·3-s − 2.20·5-s + 0.606·7-s − 2.99·9-s − 5.81·11-s − 0.556·13-s − 0.194·15-s − 17-s − 7.49·19-s + 0.0534·21-s + 5.33·23-s − 0.139·25-s − 0.527·27-s − 5.38·29-s + 3.26·31-s − 0.512·33-s − 1.33·35-s + 1.04·37-s − 0.0489·39-s + 9.39·41-s + 0.399·43-s + 6.59·45-s + 5.48·47-s − 6.63·49-s − 0.0880·51-s + 5.91·53-s + 12.8·55-s + ⋯ |
| L(s) = 1 | + 0.0508·3-s − 0.985·5-s + 0.229·7-s − 0.997·9-s − 1.75·11-s − 0.154·13-s − 0.0501·15-s − 0.242·17-s − 1.71·19-s + 0.0116·21-s + 1.11·23-s − 0.0279·25-s − 0.101·27-s − 0.999·29-s + 0.585·31-s − 0.0891·33-s − 0.226·35-s + 0.172·37-s − 0.00783·39-s + 1.46·41-s + 0.0608·43-s + 0.983·45-s + 0.800·47-s − 0.947·49-s − 0.0123·51-s + 0.812·53-s + 1.72·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6520739533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6520739533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0880T + 3T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 7 | \( 1 - 0.606T + 7T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 13 | \( 1 + 0.556T + 13T^{2} \) |
| 19 | \( 1 + 7.49T + 19T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 + 5.38T + 29T^{2} \) |
| 31 | \( 1 - 3.26T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 - 9.39T + 41T^{2} \) |
| 43 | \( 1 - 0.399T + 43T^{2} \) |
| 47 | \( 1 - 5.48T + 47T^{2} \) |
| 53 | \( 1 - 5.91T + 53T^{2} \) |
| 61 | \( 1 - 9.48T + 61T^{2} \) |
| 67 | \( 1 + 3.59T + 67T^{2} \) |
| 71 | \( 1 - 9.94T + 71T^{2} \) |
| 73 | \( 1 + 5.60T + 73T^{2} \) |
| 79 | \( 1 + 2.91T + 79T^{2} \) |
| 83 | \( 1 + 9.42T + 83T^{2} \) |
| 89 | \( 1 + 0.776T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−8.407192501259906119135000536313, −7.77761084641035940854243175410, −7.22045053774628406067414289057, −6.16052983739470195556317212099, −5.42276881442008065711637359695, −4.66990353268468019476049374633, −3.88426222472418615162749224653, −2.84215024309634286248740035761, −2.25609568147030715979921281119, −0.42902599560941825951250899145,
0.42902599560941825951250899145, 2.25609568147030715979921281119, 2.84215024309634286248740035761, 3.88426222472418615162749224653, 4.66990353268468019476049374633, 5.42276881442008065711637359695, 6.16052983739470195556317212099, 7.22045053774628406067414289057, 7.77761084641035940854243175410, 8.407192501259906119135000536313
