| L(s) = 1 | − 2-s + 4-s − 2.41·5-s − 8-s + 2.41·10-s − 11-s − 6.82·13-s + 16-s + 7.82·17-s − 4.82·19-s − 2.41·20-s + 22-s − 5.24·23-s + 0.828·25-s + 6.82·26-s + 2.82·29-s + 10.4·31-s − 32-s − 7.82·34-s − 1.65·37-s + 4.82·38-s + 2.41·40-s + 6.65·41-s + 2.82·43-s − 44-s + 5.24·46-s + 0.757·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.07·5-s − 0.353·8-s + 0.763·10-s − 0.301·11-s − 1.89·13-s + 0.250·16-s + 1.89·17-s − 1.10·19-s − 0.539·20-s + 0.213·22-s − 1.09·23-s + 0.165·25-s + 1.33·26-s + 0.525·29-s + 1.88·31-s − 0.176·32-s − 1.34·34-s − 0.272·37-s + 0.783·38-s + 0.381·40-s + 1.03·41-s + 0.431·43-s − 0.150·44-s + 0.772·46-s + 0.110·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2.41T + 5T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 17 | \( 1 - 7.82T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 + 5.24T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 - 0.757T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 + 4.41T + 61T^{2} \) |
| 67 | \( 1 - 3.48T + 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 - 0.828T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.62131354390919947494620583538, −6.89406788662219584709501861313, −6.10188572759833100261551404186, −5.27742975369708804444415983702, −4.49862789472240908485986891463, −3.81468084124013640820181280480, −2.83295431209764962518859430072, −2.25594491287096537845662703370, −0.918418781471334153838701053016, 0,
0.918418781471334153838701053016, 2.25594491287096537845662703370, 2.83295431209764962518859430072, 3.81468084124013640820181280480, 4.49862789472240908485986891463, 5.27742975369708804444415983702, 6.10188572759833100261551404186, 6.89406788662219584709501861313, 7.62131354390919947494620583538
