| L(s) = 1 | − 4·5-s − 4·11-s + 4·13-s − 4·17-s + 2·19-s − 8·23-s + 11·25-s + 6·29-s − 2·31-s − 37-s − 6·41-s − 8·43-s + 12·47-s − 2·53-s + 16·55-s + 10·59-s − 16·65-s + 12·67-s − 16·71-s + 6·73-s − 8·79-s + 4·83-s + 16·85-s − 12·89-s − 8·95-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.20·11-s + 1.10·13-s − 0.970·17-s + 0.458·19-s − 1.66·23-s + 11/5·25-s + 1.11·29-s − 0.359·31-s − 0.164·37-s − 0.937·41-s − 1.21·43-s + 1.75·47-s − 0.274·53-s + 2.15·55-s + 1.30·59-s − 1.98·65-s + 1.46·67-s − 1.89·71-s + 0.702·73-s − 0.900·79-s + 0.439·83-s + 1.73·85-s − 1.27·89-s − 0.820·95-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.84887128915909, −13.07918599875328, −12.84074630429096, −12.10393353512696, −11.81107845595080, −11.38612269266234, −10.91342641536980, −10.35178300268955, −10.12228105193203, −9.203603583391778, −8.549301081810653, −8.325375164780188, −7.988103112559492, −7.339329405918669, −6.932801880905715, −6.351284666531904, −5.680271860606208, −5.087225527041772, −4.532413924298498, −3.938321196196939, −3.664059901567652, −2.939130792719609, −2.381615559749524, −1.496959808308993, −0.6012947820706901, 0,
0.6012947820706901, 1.496959808308993, 2.381615559749524, 2.939130792719609, 3.664059901567652, 3.938321196196939, 4.532413924298498, 5.087225527041772, 5.680271860606208, 6.351284666531904, 6.932801880905715, 7.339329405918669, 7.988103112559492, 8.325375164780188, 8.549301081810653, 9.203603583391778, 10.12228105193203, 10.35178300268955, 10.91342641536980, 11.38612269266234, 11.81107845595080, 12.10393353512696, 12.84074630429096, 13.07918599875328, 13.84887128915909
