| L(s) = 1 | − 3.24·3-s − 1.49·5-s + 7.51·9-s − 2.60·11-s − 2.15·13-s + 4.85·15-s − 7.81·17-s + 4.98·19-s − 5.00·23-s − 2.75·25-s − 14.6·27-s + 0.404·29-s − 1.13·31-s + 8.43·33-s − 1.67·37-s + 7.00·39-s + 41-s − 0.0986·43-s − 11.2·45-s − 6.72·47-s + 25.3·51-s + 13.0·53-s + 3.89·55-s − 16.1·57-s − 15.1·59-s + 1.87·61-s + 3.23·65-s + ⋯ |
| L(s) = 1 | − 1.87·3-s − 0.669·5-s + 2.50·9-s − 0.784·11-s − 0.598·13-s + 1.25·15-s − 1.89·17-s + 1.14·19-s − 1.04·23-s − 0.551·25-s − 2.81·27-s + 0.0750·29-s − 0.203·31-s + 1.46·33-s − 0.275·37-s + 1.12·39-s + 0.156·41-s − 0.0150·43-s − 1.67·45-s − 0.980·47-s + 3.54·51-s + 1.78·53-s + 0.524·55-s − 2.14·57-s − 1.97·59-s + 0.239·61-s + 0.400·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08788888551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08788888551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 + 1.49T + 5T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 17 | \( 1 + 7.81T + 17T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 23 | \( 1 + 5.00T + 23T^{2} \) |
| 29 | \( 1 - 0.404T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 + 1.67T + 37T^{2} \) |
| 43 | \( 1 + 0.0986T + 43T^{2} \) |
| 47 | \( 1 + 6.72T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 - 1.87T + 61T^{2} \) |
| 67 | \( 1 + 4.60T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−7.56291359758260730168885405791, −7.08117433829839032161280994853, −6.40616251419194820169366935198, −5.68559034464528210078275693532, −5.10564414639517031843973010760, −4.45226894492959430095446173096, −3.88706251039412957899462648165, −2.56948513513773118761794364201, −1.50789591205207151099240928234, −0.16354074617591646228356762145,
0.16354074617591646228356762145, 1.50789591205207151099240928234, 2.56948513513773118761794364201, 3.88706251039412957899462648165, 4.45226894492959430095446173096, 5.10564414639517031843973010760, 5.68559034464528210078275693532, 6.40616251419194820169366935198, 7.08117433829839032161280994853, 7.56291359758260730168885405791
