| L(s) = 1 | − 3-s − 3.95·5-s − 4.62·7-s + 9-s + 3.93·11-s − 4.54·13-s + 3.95·15-s + 3.33·17-s − 7.37·19-s + 4.62·21-s + 6.16·23-s + 10.6·25-s − 27-s + 1.46·29-s − 1.63·31-s − 3.93·33-s + 18.2·35-s − 3.03·37-s + 4.54·39-s − 8.52·41-s + 11.0·43-s − 3.95·45-s − 4.29·47-s + 14.3·49-s − 3.33·51-s + 9.62·53-s − 15.5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.76·5-s − 1.74·7-s + 0.333·9-s + 1.18·11-s − 1.26·13-s + 1.02·15-s + 0.808·17-s − 1.69·19-s + 1.00·21-s + 1.28·23-s + 2.12·25-s − 0.192·27-s + 0.271·29-s − 0.293·31-s − 0.685·33-s + 3.08·35-s − 0.498·37-s + 0.727·39-s − 1.33·41-s + 1.68·43-s − 0.588·45-s − 0.626·47-s + 2.05·49-s − 0.466·51-s + 1.32·53-s − 2.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 + 3.95T + 5T^{2} \) |
| 7 | \( 1 + 4.62T + 7T^{2} \) |
| 11 | \( 1 - 3.93T + 11T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 + 7.37T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 + 3.03T + 37T^{2} \) |
| 41 | \( 1 + 8.52T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 - 9.62T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 6.58T + 67T^{2} \) |
| 71 | \( 1 + 9.75T + 71T^{2} \) |
| 73 | \( 1 + 3.29T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 6.16T + 97T^{2} \) |
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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
−7.43370999869485858295297071227, −6.92672331382261360298170849650, −6.63390924631203396081010155151, −5.61392300096765272145199031038, −4.66661353130574878138540213639, −3.93075497824793534081160278517, −3.48877499829630534940101934915, −2.54989173571968962597780328566, −0.827301795392248157509602741676, 0,
0.827301795392248157509602741676, 2.54989173571968962597780328566, 3.48877499829630534940101934915, 3.93075497824793534081160278517, 4.66661353130574878138540213639, 5.61392300096765272145199031038, 6.63390924631203396081010155151, 6.92672331382261360298170849650, 7.43370999869485858295297071227
