| L(s) = 1 | + 0.589·3-s − 1.60·5-s − 7-s − 2.65·9-s + 5.45·11-s + 6.59·13-s − 0.943·15-s + 5.33·17-s + 3.61·19-s − 0.589·21-s + 2.60·23-s − 2.43·25-s − 3.33·27-s + 1.72·29-s + 0.833·31-s + 3.21·33-s + 1.60·35-s − 6.26·37-s + 3.88·39-s − 0.263·41-s + 1.77·43-s + 4.24·45-s − 10.7·47-s + 49-s + 3.14·51-s − 0.0673·53-s − 8.73·55-s + ⋯ |
| L(s) = 1 | + 0.340·3-s − 0.715·5-s − 0.377·7-s − 0.884·9-s + 1.64·11-s + 1.82·13-s − 0.243·15-s + 1.29·17-s + 0.830·19-s − 0.128·21-s + 0.543·23-s − 0.487·25-s − 0.641·27-s + 0.321·29-s + 0.149·31-s + 0.559·33-s + 0.270·35-s − 1.02·37-s + 0.622·39-s − 0.0411·41-s + 0.270·43-s + 0.632·45-s − 1.56·47-s + 0.142·49-s + 0.440·51-s − 0.00925·53-s − 1.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.304667249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.304667249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 0.589T + 3T^{2} \) |
| 5 | \( 1 + 1.60T + 5T^{2} \) |
| 11 | \( 1 - 5.45T + 11T^{2} \) |
| 13 | \( 1 - 6.59T + 13T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 - 0.833T + 31T^{2} \) |
| 37 | \( 1 + 6.26T + 37T^{2} \) |
| 41 | \( 1 + 0.263T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 0.0673T + 53T^{2} \) |
| 59 | \( 1 - 5.10T + 59T^{2} \) |
| 61 | \( 1 + 6.31T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 2.05T + 71T^{2} \) |
| 73 | \( 1 + 5.48T + 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 + 8.25T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−8.132886858925783593041754833103, −7.25028264813022258618924405208, −6.48412075047627774603186922269, −5.95217662179261249429655974950, −5.17674870682698873677154810923, −3.98800321690427093015300094763, −3.51578139683865503198735586301, −3.11793478818137088376853311333, −1.61927935172542395438908378088, −0.814358693590518688432952063077,
0.814358693590518688432952063077, 1.61927935172542395438908378088, 3.11793478818137088376853311333, 3.51578139683865503198735586301, 3.98800321690427093015300094763, 5.17674870682698873677154810923, 5.95217662179261249429655974950, 6.48412075047627774603186922269, 7.25028264813022258618924405208, 8.132886858925783593041754833103
