| L(s) = 1 | + 3·3-s + 5-s − 5·7-s + 6·9-s + 4·11-s + 13-s + 3·15-s − 3·17-s − 19-s − 15·21-s + 7·23-s + 25-s + 9·27-s + 3·29-s − 2·31-s + 12·33-s − 5·35-s + 2·37-s + 3·39-s − 6·41-s − 6·43-s + 6·45-s + 18·49-s − 9·51-s + 13·53-s + 4·55-s − 3·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 1.88·7-s + 2·9-s + 1.20·11-s + 0.277·13-s + 0.774·15-s − 0.727·17-s − 0.229·19-s − 3.27·21-s + 1.45·23-s + 1/5·25-s + 1.73·27-s + 0.557·29-s − 0.359·31-s + 2.08·33-s − 0.845·35-s + 0.328·37-s + 0.480·39-s − 0.937·41-s − 0.914·43-s + 0.894·45-s + 18/7·49-s − 1.26·51-s + 1.78·53-s + 0.539·55-s − 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.857425987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.857425987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.377037533232944812105183539894, −7.17565191181093040081765701646, −6.76779382257262786041789222840, −6.32305305518926285004104181420, −5.12093453824991032563935879013, −3.90110139741174767292876467810, −3.62982038370431966424747728455, −2.80805664330270782994541647617, −2.16844846852651330874505340255, −0.966764407739515887382742224218,
0.966764407739515887382742224218, 2.16844846852651330874505340255, 2.80805664330270782994541647617, 3.62982038370431966424747728455, 3.90110139741174767292876467810, 5.12093453824991032563935879013, 6.32305305518926285004104181420, 6.76779382257262786041789222840, 7.17565191181093040081765701646, 8.377037533232944812105183539894
