L(s) = 1 | − 3-s + 9-s − 11-s − 13-s − 4·17-s + 8·23-s − 5·25-s − 27-s + 10·31-s + 33-s − 2·37-s + 39-s − 6·41-s + 2·43-s − 4·47-s − 7·49-s + 4·51-s + 14·53-s + 6·61-s − 6·67-s − 8·69-s − 8·71-s − 6·73-s + 5·75-s − 2·79-s + 81-s + 12·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.970·17-s + 1.66·23-s − 25-s − 0.192·27-s + 1.79·31-s + 0.174·33-s − 0.328·37-s + 0.160·39-s − 0.937·41-s + 0.304·43-s − 0.583·47-s − 49-s + 0.560·51-s + 1.92·53-s + 0.768·61-s − 0.733·67-s − 0.963·69-s − 0.949·71-s − 0.702·73-s + 0.577·75-s − 0.225·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.48293834090569897573536615041, −6.85113001676360005293782761991, −6.29331026780807884135216723441, −5.41879309435301368918323829105, −4.82227119072088025634882057980, −4.15621683268257127526849089343, −3.11151244898854865968917192141, −2.29115177582657524553545579984, −1.18015209720507627245618132504, 0,
1.18015209720507627245618132504, 2.29115177582657524553545579984, 3.11151244898854865968917192141, 4.15621683268257127526849089343, 4.82227119072088025634882057980, 5.41879309435301368918323829105, 6.29331026780807884135216723441, 6.85113001676360005293782761991, 7.48293834090569897573536615041