L(s) = 1 | − 3-s − 7-s + 9-s − 5·13-s − 4·17-s + 3·19-s + 21-s − 27-s − 7·29-s + 7·31-s − 8·37-s + 5·39-s + 41-s − 43-s + 6·47-s − 6·49-s + 4·51-s − 2·53-s − 3·57-s − 5·61-s − 63-s + 11·67-s + 6·71-s + 5·73-s + 79-s + 81-s − 8·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.38·13-s − 0.970·17-s + 0.688·19-s + 0.218·21-s − 0.192·27-s − 1.29·29-s + 1.25·31-s − 1.31·37-s + 0.800·39-s + 0.156·41-s − 0.152·43-s + 0.875·47-s − 6/7·49-s + 0.560·51-s − 0.274·53-s − 0.397·57-s − 0.640·61-s − 0.125·63-s + 1.34·67-s + 0.712·71-s + 0.585·73-s + 0.112·79-s + 1/9·81-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.20825318613220, −12.73190634403303, −12.26251783433719, −11.95160633955506, −11.40900049234444, −10.94762583295104, −10.52573444663286, −9.871023423909229, −9.595571220225835, −9.226348804665054, −8.485373482715709, −8.033266267042357, −7.390835595254652, −6.989172966224035, −6.648098354876050, −6.020395994153846, −5.405630473084113, −5.085181435533301, −4.494955499366396, −4.002002333927528, −3.308479911243548, −2.718688129730887, −2.134516770485121, −1.518899241829192, −0.6008048051040776, 0,
0.6008048051040776, 1.518899241829192, 2.134516770485121, 2.718688129730887, 3.308479911243548, 4.002002333927528, 4.494955499366396, 5.085181435533301, 5.405630473084113, 6.020395994153846, 6.648098354876050, 6.989172966224035, 7.390835595254652, 8.033266267042357, 8.485373482715709, 9.226348804665054, 9.595571220225835, 9.871023423909229, 10.52573444663286, 10.94762583295104, 11.40900049234444, 11.95160633955506, 12.26251783433719, 12.73190634403303, 13.20825318613220