| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 4·11-s + 15-s − 17-s − 2·21-s + 8·23-s + 25-s − 27-s + 6·29-s + 2·31-s + 4·33-s − 2·35-s − 4·37-s + 2·41-s − 4·43-s − 45-s − 2·47-s − 3·49-s + 51-s + 10·53-s + 4·55-s − 10·59-s + 14·61-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.258·15-s − 0.242·17-s − 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.696·33-s − 0.338·35-s − 0.657·37-s + 0.312·41-s − 0.609·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s + 0.140·51-s + 1.37·53-s + 0.539·55-s − 1.30·59-s + 1.79·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.76412619616620, −12.34667015869811, −11.77277308801446, −11.38487706143194, −11.01993595924644, −10.68015006885656, −10.04763886285948, −9.885051657317842, −9.003015901750379, −8.529477803154422, −8.309431162887608, −7.672927614291067, −7.246655256433390, −6.835987128412480, −6.323924438763556, −5.656738499593743, −5.096390234331116, −4.918474664154396, −4.470023153448430, −3.738372604848966, −3.175663260411863, −2.591318253712137, −2.081214103125981, −1.233243182533607, −0.7689387878402854, 0,
0.7689387878402854, 1.233243182533607, 2.081214103125981, 2.591318253712137, 3.175663260411863, 3.738372604848966, 4.470023153448430, 4.918474664154396, 5.096390234331116, 5.656738499593743, 6.323924438763556, 6.835987128412480, 7.246655256433390, 7.672927614291067, 8.309431162887608, 8.529477803154422, 9.003015901750379, 9.885051657317842, 10.04763886285948, 10.68015006885656, 11.01993595924644, 11.38487706143194, 11.77277308801446, 12.34667015869811, 12.76412619616620
