L(s) = 1 | + 2·5-s − 2·11-s − 13-s + 4·19-s + 4·23-s − 25-s − 8·29-s − 8·31-s − 2·37-s − 6·41-s − 4·43-s + 6·47-s − 7·49-s − 4·53-s − 4·55-s − 6·59-s − 2·61-s − 2·65-s + 4·67-s + 6·71-s − 2·73-s + 16·79-s − 2·83-s − 10·89-s + 8·95-s − 2·97-s + 8·101-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.603·11-s − 0.277·13-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.48·29-s − 1.43·31-s − 0.328·37-s − 0.937·41-s − 0.609·43-s + 0.875·47-s − 49-s − 0.549·53-s − 0.539·55-s − 0.781·59-s − 0.256·61-s − 0.248·65-s + 0.488·67-s + 0.712·71-s − 0.234·73-s + 1.80·79-s − 0.219·83-s − 1.05·89-s + 0.820·95-s − 0.203·97-s + 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.50791422201769086894669894726, −6.90114405359056576294707238062, −6.06613226153229835334011663550, −5.29954661806540540403915329863, −5.06912568895291456570565150351, −3.81449478526603532182762479142, −3.10690762055903884208672003073, −2.16585700427808944425817318448, −1.45425355028563710334297526724, 0,
1.45425355028563710334297526724, 2.16585700427808944425817318448, 3.10690762055903884208672003073, 3.81449478526603532182762479142, 5.06912568895291456570565150351, 5.29954661806540540403915329863, 6.06613226153229835334011663550, 6.90114405359056576294707238062, 7.50791422201769086894669894726