| L(s) = 1 | − 5.86·3-s + 4.13·5-s − 6.13·7-s + 7.40·9-s − 15.7·11-s + 13·13-s − 24.2·15-s − 7.86·17-s + 94.6·19-s + 35.9·21-s + 164.·23-s − 107.·25-s + 114.·27-s + 146.·29-s + 113.·31-s + 92.2·33-s − 25.3·35-s + 295.·37-s − 76.2·39-s + 356.·41-s + 197.·43-s + 30.6·45-s − 308.·47-s − 305.·49-s + 46.1·51-s + 84.9·53-s − 65.0·55-s + ⋯ |
| L(s) = 1 | − 1.12·3-s + 0.369·5-s − 0.331·7-s + 0.274·9-s − 0.431·11-s + 0.277·13-s − 0.417·15-s − 0.112·17-s + 1.14·19-s + 0.373·21-s + 1.49·23-s − 0.863·25-s + 0.819·27-s + 0.939·29-s + 0.658·31-s + 0.486·33-s − 0.122·35-s + 1.31·37-s − 0.313·39-s + 1.35·41-s + 0.701·43-s + 0.101·45-s − 0.957·47-s − 0.890·49-s + 0.126·51-s + 0.220·53-s − 0.159·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.122432120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122432120\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 + 5.86T + 27T^{2} \) |
| 5 | \( 1 - 4.13T + 125T^{2} \) |
| 7 | \( 1 + 6.13T + 343T^{2} \) |
| 11 | \( 1 + 15.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 7.86T + 4.91e3T^{2} \) |
| 19 | \( 1 - 94.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 146.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 295.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 356.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 308.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 84.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 293.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 416.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 186.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 588.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 582.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 872.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 767.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
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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
−11.78189567422102239709755852019, −11.08126012739548310763030088101, −10.12596668510281634922839581902, −9.159357773657733976594203501833, −7.76508360054528830943903312562, −6.52401203327968616107780654326, −5.70900202104630958889974453955, −4.70832427721815254519493718895, −2.91534292769774742221777575665, −0.860053429496567177115096604445,
0.860053429496567177115096604445, 2.91534292769774742221777575665, 4.70832427721815254519493718895, 5.70900202104630958889974453955, 6.52401203327968616107780654326, 7.76508360054528830943903312562, 9.159357773657733976594203501833, 10.12596668510281634922839581902, 11.08126012739548310763030088101, 11.78189567422102239709755852019
