| L(s) = 1 | − 3-s − 2·4-s + 5-s − 7-s + 9-s − 2·11-s + 2·12-s − 15-s + 4·16-s + 17-s − 2·19-s − 2·20-s + 21-s − 23-s + 25-s − 27-s + 2·28-s + 8·29-s − 31-s + 2·33-s − 35-s − 2·36-s + 3·37-s + 7·41-s + 4·44-s + 45-s + 47-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 0.258·15-s + 16-s + 0.242·17-s − 0.458·19-s − 0.447·20-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.377·28-s + 1.48·29-s − 0.179·31-s + 0.348·33-s − 0.169·35-s − 1/3·36-s + 0.493·37-s + 1.09·41-s + 0.603·44-s + 0.149·45-s + 0.145·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.89606725471356, −12.52173474590320, −12.12936883253040, −11.64512069894254, −10.86343183363074, −10.54812522684612, −10.28377942455691, −9.572433571224716, −9.403230413804846, −8.877071548885869, −8.223141228398340, −7.910817277743213, −7.390996492546571, −6.682523630647175, −6.199508565019585, −5.888661481417929, −5.300529994754967, −4.821930704754332, −4.442821601753852, −3.902785150968876, −3.194281326088922, −2.744362829028086, −2.021701051817524, −1.241017783795467, −0.6781787179759439, 0,
0.6781787179759439, 1.241017783795467, 2.021701051817524, 2.744362829028086, 3.194281326088922, 3.902785150968876, 4.442821601753852, 4.821930704754332, 5.300529994754967, 5.888661481417929, 6.199508565019585, 6.682523630647175, 7.390996492546571, 7.910817277743213, 8.223141228398340, 8.877071548885869, 9.403230413804846, 9.572433571224716, 10.28377942455691, 10.54812522684612, 10.86343183363074, 11.64512069894254, 12.12936883253040, 12.52173474590320, 12.89606725471356
