| L(s) = 1 | − 0.579·3-s + 2.10·5-s − 2.73·7-s − 2.66·9-s + 4.66·11-s + 4.47·13-s − 1.22·15-s − 6.85·17-s + 19-s + 1.58·21-s − 1.20·23-s − 0.552·25-s + 3.27·27-s − 9.57·29-s + 5.35·31-s − 2.70·33-s − 5.76·35-s − 1.09·37-s − 2.59·39-s − 7.33·41-s + 7.64·43-s − 5.61·45-s − 7.56·47-s + 0.476·49-s + 3.96·51-s − 3.11·53-s + 9.83·55-s + ⋯ |
| L(s) = 1 | − 0.334·3-s + 0.943·5-s − 1.03·7-s − 0.888·9-s + 1.40·11-s + 1.24·13-s − 0.315·15-s − 1.66·17-s + 0.229·19-s + 0.345·21-s − 0.251·23-s − 0.110·25-s + 0.631·27-s − 1.77·29-s + 0.962·31-s − 0.470·33-s − 0.974·35-s − 0.180·37-s − 0.415·39-s − 1.14·41-s + 1.16·43-s − 0.837·45-s − 1.10·47-s + 0.0681·49-s + 0.555·51-s − 0.428·53-s + 1.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.579T + 3T^{2} \) |
| 5 | \( 1 - 2.10T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 4.66T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 6.85T + 17T^{2} \) |
| 23 | \( 1 + 1.20T + 23T^{2} \) |
| 29 | \( 1 + 9.57T + 29T^{2} \) |
| 31 | \( 1 - 5.35T + 31T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 + 7.33T + 41T^{2} \) |
| 43 | \( 1 - 7.64T + 43T^{2} \) |
| 47 | \( 1 + 7.56T + 47T^{2} \) |
| 53 | \( 1 + 3.11T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 0.722T + 61T^{2} \) |
| 67 | \( 1 - 6.13T + 67T^{2} \) |
| 71 | \( 1 - 4.62T + 71T^{2} \) |
| 73 | \( 1 - 6.19T + 73T^{2} \) |
| 79 | \( 1 - 3.26T + 79T^{2} \) |
| 83 | \( 1 - 8.97T + 83T^{2} \) |
| 89 | \( 1 + 0.620T + 89T^{2} \) |
| 97 | \( 1 + 1.67T + 97T^{2} \) |
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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
−8.029755002873982349211941543668, −6.71996133063030314184562980494, −6.40054725178909679952443654574, −6.00572669802067354824967821171, −5.13446997765639051834593513276, −4.00826155718859093051763719679, −3.40685241240796622729155592187, −2.33790566449017975380410376431, −1.40589205533259423042495015203, 0,
1.40589205533259423042495015203, 2.33790566449017975380410376431, 3.40685241240796622729155592187, 4.00826155718859093051763719679, 5.13446997765639051834593513276, 6.00572669802067354824967821171, 6.40054725178909679952443654574, 6.71996133063030314184562980494, 8.029755002873982349211941543668
