| L(s) = 1 | − 3-s + 7-s + 9-s − 3·11-s − 3·13-s − 5·19-s − 21-s + 4·23-s − 27-s + 5·29-s − 2·31-s + 3·33-s − 10·37-s + 3·39-s − 43-s + 3·47-s − 6·49-s + 4·53-s + 5·57-s − 4·59-s + 8·61-s + 63-s + 2·67-s − 4·69-s + 16·73-s − 3·77-s − 14·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.832·13-s − 1.14·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s + 0.928·29-s − 0.359·31-s + 0.522·33-s − 1.64·37-s + 0.480·39-s − 0.152·43-s + 0.437·47-s − 6/7·49-s + 0.549·53-s + 0.662·57-s − 0.520·59-s + 1.02·61-s + 0.125·63-s + 0.244·67-s − 0.481·69-s + 1.87·73-s − 0.341·77-s − 1.57·79-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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.04894737126162, −12.74907836761768, −12.43917752543971, −11.80562792235467, −11.44898767118156, −10.78260098295344, −10.60185183032990, −10.06053186596093, −9.687443699406211, −8.838114920065683, −8.639161800798491, −7.990852753144831, −7.531564860087617, −6.956004276906018, −6.647519958356586, −5.989484202695715, −5.377912098369619, −4.971991873679488, −4.681870432781981, −3.967185350227693, −3.339288314297787, −2.629059491966652, −2.161630985217053, −1.515004601998864, −0.6543539670988740, 0,
0.6543539670988740, 1.515004601998864, 2.161630985217053, 2.629059491966652, 3.339288314297787, 3.967185350227693, 4.681870432781981, 4.971991873679488, 5.377912098369619, 5.989484202695715, 6.647519958356586, 6.956004276906018, 7.531564860087617, 7.990852753144831, 8.639161800798491, 8.838114920065683, 9.687443699406211, 10.06053186596093, 10.60185183032990, 10.78260098295344, 11.44898767118156, 11.80562792235467, 12.43917752543971, 12.74907836761768, 13.04894737126162
