| L(s) = 1 | + 2·5-s + 7-s − 3·9-s − 2·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s + 8·31-s + 2·35-s − 2·37-s − 2·41-s + 4·43-s − 6·45-s − 8·47-s + 49-s + 6·53-s + 6·61-s − 3·63-s − 4·65-s − 4·67-s − 8·71-s − 10·73-s − 16·79-s + 9·81-s − 8·83-s + 12·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 9-s − 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s − 0.312·41-s + 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.768·61-s − 0.377·63-s − 0.496·65-s − 0.488·67-s − 0.949·71-s − 1.17·73-s − 1.80·79-s + 81-s − 0.878·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.69587064441186150768078543731, −6.88310002274215763839569601337, −5.94974241082854154889853475745, −5.72315905059004231786170242312, −4.86884684581528043387154729760, −4.03151585237256402081332864489, −2.98780927444278559643957768594, −2.30074942792092931708066021417, −1.44245425010921288807179765849, 0,
1.44245425010921288807179765849, 2.30074942792092931708066021417, 2.98780927444278559643957768594, 4.03151585237256402081332864489, 4.86884684581528043387154729760, 5.72315905059004231786170242312, 5.94974241082854154889853475745, 6.88310002274215763839569601337, 7.69587064441186150768078543731
