| L(s) = 1 | − 2.23·3-s − 5-s − 2·7-s + 2.00·9-s + 4.47·13-s + 2.23·15-s − 4.47·17-s + 6·19-s + 4.47·21-s + 2.23·23-s − 4·25-s + 2.23·27-s − 8.94·29-s + 2.23·31-s + 2·35-s − 7·37-s − 10.0·39-s − 4.47·41-s − 4·43-s − 2.00·45-s − 8.94·47-s − 3·49-s + 10.0·51-s + 6·53-s − 13.4·57-s + 11.1·59-s + 8.94·61-s + ⋯ |
| L(s) = 1 | − 1.29·3-s − 0.447·5-s − 0.755·7-s + 0.666·9-s + 1.24·13-s + 0.577·15-s − 1.08·17-s + 1.37·19-s + 0.975·21-s + 0.466·23-s − 0.800·25-s + 0.430·27-s − 1.66·29-s + 0.401·31-s + 0.338·35-s − 1.15·37-s − 1.60·39-s − 0.698·41-s − 0.609·43-s − 0.298·45-s − 1.30·47-s − 0.428·49-s + 1.40·51-s + 0.824·53-s − 1.77·57-s + 1.45·59-s + 1.14·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 2.23T + 23T^{2} \) |
| 29 | \( 1 + 8.94T + 29T^{2} \) |
| 31 | \( 1 - 2.23T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 8.94T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 6.70T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
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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
−7.28611596361409588870020032068, −6.62959227496604694603637158077, −6.25118139849277202438466544731, −5.33066739748235138209750570993, −5.01310558201113139047297274281, −3.64684345307078130715520655152, −3.58241100535013479123316662261, −2.11038101479190193356541012198, −0.933007966542278322497195998809, 0,
0.933007966542278322497195998809, 2.11038101479190193356541012198, 3.58241100535013479123316662261, 3.64684345307078130715520655152, 5.01310558201113139047297274281, 5.33066739748235138209750570993, 6.25118139849277202438466544731, 6.62959227496604694603637158077, 7.28611596361409588870020032068
