| L(s) = 1 | − 3-s − 6·7-s − 26·9-s + 19·11-s − 12·13-s − 75·17-s + 91·19-s + 6·21-s + 174·23-s + 53·27-s + 272·29-s − 230·31-s − 19·33-s + 182·37-s + 12·39-s + 117·41-s − 372·43-s − 52·47-s − 307·49-s + 75·51-s + 402·53-s − 91·57-s − 312·59-s − 170·61-s + 156·63-s − 763·67-s − 174·69-s + ⋯ |
| L(s) = 1 | − 0.192·3-s − 0.323·7-s − 0.962·9-s + 0.520·11-s − 0.256·13-s − 1.07·17-s + 1.09·19-s + 0.0623·21-s + 1.57·23-s + 0.377·27-s + 1.74·29-s − 1.33·31-s − 0.100·33-s + 0.808·37-s + 0.0492·39-s + 0.445·41-s − 1.31·43-s − 0.161·47-s − 0.895·49-s + 0.205·51-s + 1.04·53-s − 0.211·57-s − 0.688·59-s − 0.356·61-s + 0.311·63-s − 1.39·67-s − 0.303·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 19 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 75 T + p^{3} T^{2} \) |
| 19 | \( 1 - 91 T + p^{3} T^{2} \) |
| 23 | \( 1 - 174 T + p^{3} T^{2} \) |
| 29 | \( 1 - 272 T + p^{3} T^{2} \) |
| 31 | \( 1 + 230 T + p^{3} T^{2} \) |
| 37 | \( 1 - 182 T + p^{3} T^{2} \) |
| 41 | \( 1 - 117 T + p^{3} T^{2} \) |
| 43 | \( 1 + 372 T + p^{3} T^{2} \) |
| 47 | \( 1 + 52 T + p^{3} T^{2} \) |
| 53 | \( 1 - 402 T + p^{3} T^{2} \) |
| 59 | \( 1 + 312 T + p^{3} T^{2} \) |
| 61 | \( 1 + 170 T + p^{3} T^{2} \) |
| 67 | \( 1 + 763 T + p^{3} T^{2} \) |
| 71 | \( 1 + 52 T + p^{3} T^{2} \) |
| 73 | \( 1 + 981 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1054 T + p^{3} T^{2} \) |
| 83 | \( 1 + 351 T + p^{3} T^{2} \) |
| 89 | \( 1 - 799 T + p^{3} T^{2} \) |
| 97 | \( 1 - 962 T + p^{3} T^{2} \) |
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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
−8.896561822406311441111753145176, −7.84380581173740693194605664487, −6.90572190646516369979060593140, −6.30011943221810946942119212820, −5.31708969802416174907580591236, −4.58842911362047858826839861196, −3.34034787835215157662841224525, −2.63363316434026053250358179212, −1.19810071444731994185278218094, 0,
1.19810071444731994185278218094, 2.63363316434026053250358179212, 3.34034787835215157662841224525, 4.58842911362047858826839861196, 5.31708969802416174907580591236, 6.30011943221810946942119212820, 6.90572190646516369979060593140, 7.84380581173740693194605664487, 8.896561822406311441111753145176
