| L(s) = 1 | + 3-s + 5-s + 9-s + 4·11-s + 15-s − 17-s − 6·19-s + 4·23-s + 25-s + 27-s − 6·29-s + 4·31-s + 4·33-s + 6·37-s + 2·41-s − 4·43-s + 45-s − 2·47-s − 7·49-s − 51-s − 6·53-s + 4·55-s − 6·57-s + 10·59-s − 2·61-s + 6·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.258·15-s − 0.242·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s + 0.986·37-s + 0.312·41-s − 0.609·43-s + 0.149·45-s − 0.291·47-s − 49-s − 0.140·51-s − 0.824·53-s + 0.539·55-s − 0.794·57-s + 1.30·59-s − 0.256·61-s + 0.733·67-s + 0.481·69-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
| \(L(1)\) |
\(\approx\) |
\(4.164631724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.164631724\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.64020555406029, −12.22221650683280, −11.56789913979025, −11.13333217167358, −10.89117854183520, −10.17532221547350, −9.670173280945511, −9.436321871852002, −8.922986130038432, −8.503172579853842, −8.068545797716248, −7.552591245138554, −6.787470526118940, −6.640825157250412, −6.186060537544002, −5.572553087377397, −4.896338788084991, −4.501975993061706, −3.911455038331088, −3.522990750263863, −2.864563243081467, −2.254106133788752, −1.827895638926886, −1.213764316618406, −0.5162121916260201,
0.5162121916260201, 1.213764316618406, 1.827895638926886, 2.254106133788752, 2.864563243081467, 3.522990750263863, 3.911455038331088, 4.501975993061706, 4.896338788084991, 5.572553087377397, 6.186060537544002, 6.640825157250412, 6.787470526118940, 7.552591245138554, 8.068545797716248, 8.503172579853842, 8.922986130038432, 9.436321871852002, 9.670173280945511, 10.17532221547350, 10.89117854183520, 11.13333217167358, 11.56789913979025, 12.22221650683280, 12.64020555406029
