| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s − 2·11-s + 13-s − 15-s − 17-s − 6·19-s + 2·21-s + 2·23-s + 25-s − 27-s + 3·29-s + 31-s + 2·33-s − 2·35-s − 8·37-s − 39-s + 10·41-s + 2·43-s + 45-s + 8·47-s − 3·49-s + 51-s + 13·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s − 1.37·19-s + 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 0.179·31-s + 0.348·33-s − 0.338·35-s − 1.31·37-s − 0.160·39-s + 1.56·41-s + 0.304·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.140·51-s + 1.78·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.134980992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.134980992\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.36507760271976, −14.89194720824259, −14.10733190823544, −13.62804410603390, −13.01802657424009, −12.70898167476722, −12.20096977841969, −11.47925963399153, −10.84282536321896, −10.39491955628130, −10.08648768727593, −9.231640079049080, −8.802415629891543, −8.209770118306507, −7.314665160929803, −6.864466148846276, −6.267303733586588, −5.743989246124263, −5.223486154789112, −4.351165459437814, −3.918065538042638, −2.856161623652413, −2.405814187972654, −1.415918604849102, −0.4388912061820520,
0.4388912061820520, 1.415918604849102, 2.405814187972654, 2.856161623652413, 3.918065538042638, 4.351165459437814, 5.223486154789112, 5.743989246124263, 6.267303733586588, 6.864466148846276, 7.314665160929803, 8.209770118306507, 8.802415629891543, 9.231640079049080, 10.08648768727593, 10.39491955628130, 10.84282536321896, 11.47925963399153, 12.20096977841969, 12.70898167476722, 13.01802657424009, 13.62804410603390, 14.10733190823544, 14.89194720824259, 15.36507760271976
