| L(s) = 1 | + 2.03·2-s + 3.03·3-s + 2.12·4-s − 3.82·5-s + 6.15·6-s + 7-s + 0.255·8-s + 6.18·9-s − 7.77·10-s − 5.96·11-s + 6.44·12-s + 1.44·13-s + 2.03·14-s − 11.6·15-s − 3.73·16-s + 6.06·17-s + 12.5·18-s − 0.0743·19-s − 8.13·20-s + 3.03·21-s − 12.1·22-s − 4.43·23-s + 0.774·24-s + 9.64·25-s + 2.93·26-s + 9.66·27-s + 2.12·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.75·3-s + 1.06·4-s − 1.71·5-s + 2.51·6-s + 0.377·7-s + 0.0903·8-s + 2.06·9-s − 2.45·10-s − 1.79·11-s + 1.86·12-s + 0.400·13-s + 0.542·14-s − 2.99·15-s − 0.933·16-s + 1.47·17-s + 2.96·18-s − 0.0170·19-s − 1.81·20-s + 0.661·21-s − 2.58·22-s − 0.924·23-s + 0.158·24-s + 1.92·25-s + 0.575·26-s + 1.85·27-s + 0.401·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.263200634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.263200634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.03T + 2T^{2} \) |
| 3 | \( 1 - 3.03T + 3T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 11 | \( 1 + 5.96T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 - 6.06T + 17T^{2} \) |
| 19 | \( 1 + 0.0743T + 19T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 - 1.76T + 31T^{2} \) |
| 37 | \( 1 - 0.497T + 37T^{2} \) |
| 43 | \( 1 - 4.10T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 2.94T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 - 5.87T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 0.670T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−12.23874543179841311145444398384, −11.18737990227909115945640392991, −9.982946136342154827411715872472, −8.448362815371434915124336360063, −7.973066262046866038947914985214, −7.24740242067714076184156377707, −5.35292828495265709738292240783, −4.19461918768984561879924078899, −3.49343192782241138482245813336, −2.61990715011170594199354101007,
2.61990715011170594199354101007, 3.49343192782241138482245813336, 4.19461918768984561879924078899, 5.35292828495265709738292240783, 7.24740242067714076184156377707, 7.973066262046866038947914985214, 8.448362815371434915124336360063, 9.982946136342154827411715872472, 11.18737990227909115945640392991, 12.23874543179841311145444398384
