| L(s) = 1 | + 3-s + 3.46·5-s − 1.73·7-s − 2·9-s − 5.19·13-s + 3.46·15-s − 3·17-s − 19-s − 1.73·21-s + 1.73·23-s + 6.99·25-s − 5·27-s − 1.73·29-s − 3.46·31-s − 5.99·35-s − 5.19·39-s + 10·43-s − 6.92·45-s + 10.3·47-s − 4·49-s − 3·51-s − 5.19·53-s − 57-s − 9·59-s − 10.3·61-s + 3.46·63-s − 18·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.54·5-s − 0.654·7-s − 0.666·9-s − 1.44·13-s + 0.894·15-s − 0.727·17-s − 0.229·19-s − 0.377·21-s + 0.361·23-s + 1.39·25-s − 0.962·27-s − 0.321·29-s − 0.622·31-s − 1.01·35-s − 0.832·39-s + 1.52·43-s − 1.03·45-s + 1.51·47-s − 0.571·49-s − 0.420·51-s − 0.713·53-s − 0.132·57-s − 1.17·59-s − 1.33·61-s + 0.436·63-s − 2.23·65-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 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−7.88560844676676645445875268644, −7.17233727220419736760226530077, −6.37247181868020728973557357058, −5.77740366318043886860476785735, −5.11352737074415760587366585072, −4.15594723605491411798983175708, −2.89639130799549490579714737447, −2.55689286683143414712185358506, −1.68935833269533697548917956700, 0,
1.68935833269533697548917956700, 2.55689286683143414712185358506, 2.89639130799549490579714737447, 4.15594723605491411798983175708, 5.11352737074415760587366585072, 5.77740366318043886860476785735, 6.37247181868020728973557357058, 7.17233727220419736760226530077, 7.88560844676676645445875268644
