| L(s) = 1 | − 2·4-s + 7-s + 5·13-s + 4·16-s − 7·19-s − 5·25-s − 2·28-s + 4·31-s − 11·37-s + 8·43-s − 6·49-s − 10·52-s + 61-s − 8·64-s + 5·67-s + 7·73-s + 14·76-s − 17·79-s + 5·91-s + 19·97-s + 10·100-s − 13·103-s − 2·109-s + 4·112-s + ⋯ |
| L(s) = 1 | − 4-s + 0.377·7-s + 1.38·13-s + 16-s − 1.60·19-s − 25-s − 0.377·28-s + 0.718·31-s − 1.80·37-s + 1.21·43-s − 6/7·49-s − 1.38·52-s + 0.128·61-s − 64-s + 0.610·67-s + 0.819·73-s + 1.60·76-s − 1.91·79-s + 0.524·91-s + 1.92·97-s + 100-s − 1.28·103-s − 0.191·109-s + 0.377·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−7.73555418303022546350474382284, −6.71007189646149044087785786857, −6.05261256470973018797384585073, −5.41879491955347593603656581049, −4.55135766529264807402906794900, −4.00653799444701003240685090593, −3.35759065619796747743550815545, −2.11720214723806627370982255410, −1.19921815997042263362560685425, 0,
1.19921815997042263362560685425, 2.11720214723806627370982255410, 3.35759065619796747743550815545, 4.00653799444701003240685090593, 4.55135766529264807402906794900, 5.41879491955347593603656581049, 6.05261256470973018797384585073, 6.71007189646149044087785786857, 7.73555418303022546350474382284
