| L(s) = 1 | − 2·3-s − 5·5-s − 6·7-s − 23·9-s − 32·11-s − 38·13-s + 10·15-s + 26·17-s − 100·19-s + 12·21-s + 78·23-s + 25·25-s + 100·27-s − 50·29-s + 108·31-s + 64·33-s + 30·35-s + 266·37-s + 76·39-s + 22·41-s − 442·43-s + 115·45-s + 514·47-s − 307·49-s − 52·51-s + 2·53-s + 160·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.447·5-s − 0.323·7-s − 0.851·9-s − 0.877·11-s − 0.810·13-s + 0.172·15-s + 0.370·17-s − 1.20·19-s + 0.124·21-s + 0.707·23-s + 1/5·25-s + 0.712·27-s − 0.320·29-s + 0.625·31-s + 0.337·33-s + 0.144·35-s + 1.18·37-s + 0.312·39-s + 0.0838·41-s − 1.56·43-s + 0.380·45-s + 1.59·47-s − 0.895·49-s − 0.142·51-s + 0.00518·53-s + 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 26 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 78 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 266 T + p^{3} T^{2} \) |
| 41 | \( 1 - 22 T + p^{3} T^{2} \) |
| 43 | \( 1 + 442 T + p^{3} T^{2} \) |
| 47 | \( 1 - 514 T + p^{3} T^{2} \) |
| 53 | \( 1 - 2 T + p^{3} T^{2} \) |
| 59 | \( 1 + 500 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 126 T + p^{3} T^{2} \) |
| 71 | \( 1 + 412 T + p^{3} T^{2} \) |
| 73 | \( 1 + 878 T + p^{3} T^{2} \) |
| 79 | \( 1 + 600 T + p^{3} T^{2} \) |
| 83 | \( 1 + 282 T + p^{3} T^{2} \) |
| 89 | \( 1 + 150 T + p^{3} T^{2} \) |
| 97 | \( 1 - 386 T + p^{3} T^{2} \) |
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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
−13.19916963121842390903970468399, −12.24010618878904062885601772795, −11.18908905957269009262159067636, −10.16888674163776461427892641324, −8.721601056253605451008279676098, −7.54163429329196184523456068222, −6.09808728439087014876801627768, −4.75802322654636899149247552097, −2.85631746265120405741129055688, 0,
2.85631746265120405741129055688, 4.75802322654636899149247552097, 6.09808728439087014876801627768, 7.54163429329196184523456068222, 8.721601056253605451008279676098, 10.16888674163776461427892641324, 11.18908905957269009262159067636, 12.24010618878904062885601772795, 13.19916963121842390903970468399
