| L(s) = 1 | + 2.47·2-s − 3-s + 4.12·4-s + 2.79·5-s − 2.47·6-s + 5.26·8-s + 9-s + 6.91·10-s + 3.78·11-s − 4.12·12-s + 2.95·13-s − 2.79·15-s + 4.78·16-s + 3.46·17-s + 2.47·18-s − 19-s + 11.5·20-s + 9.37·22-s − 3.78·23-s − 5.26·24-s + 2.80·25-s + 7.30·26-s − 27-s − 6.57·29-s − 6.91·30-s − 5.78·31-s + 1.31·32-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.577·3-s + 2.06·4-s + 1.24·5-s − 1.01·6-s + 1.86·8-s + 0.333·9-s + 2.18·10-s + 1.14·11-s − 1.19·12-s + 0.818·13-s − 0.721·15-s + 1.19·16-s + 0.840·17-s + 0.583·18-s − 0.229·19-s + 2.57·20-s + 1.99·22-s − 0.789·23-s − 1.07·24-s + 0.560·25-s + 1.43·26-s − 0.192·27-s − 1.22·29-s − 1.26·30-s − 1.03·31-s + 0.231·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.113473371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.113473371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 - 2.95T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 + 3.78T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 + 9.84T + 37T^{2} \) |
| 41 | \( 1 + 1.30T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 6.99T + 47T^{2} \) |
| 53 | \( 1 + 4.63T + 53T^{2} \) |
| 59 | \( 1 + 7.57T + 59T^{2} \) |
| 61 | \( 1 + 2.67T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 5.64T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 8.65T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−9.012239807687043018661461033182, −7.65441848789922356211055434480, −6.76837014961313404136015090587, −6.15481739611421379557893109969, −5.69994719254141118026368688315, −5.12467872963954837388906052874, −3.97855747662709812357323933511, −3.54669920516011262012917849672, −2.17215234624370782975282065916, −1.46369122451148448283734698556,
1.46369122451148448283734698556, 2.17215234624370782975282065916, 3.54669920516011262012917849672, 3.97855747662709812357323933511, 5.12467872963954837388906052874, 5.69994719254141118026368688315, 6.15481739611421379557893109969, 6.76837014961313404136015090587, 7.65441848789922356211055434480, 9.012239807687043018661461033182
