| L(s) = 1 | + 20·5-s − 44·11-s − 44·13-s − 72·17-s + 100·19-s + 120·23-s + 275·25-s − 218·29-s − 280·31-s − 30·37-s − 120·41-s + 220·43-s − 88·47-s − 110·53-s − 880·55-s − 580·59-s + 380·61-s − 880·65-s − 980·67-s + 112·71-s − 640·73-s − 488·79-s − 660·83-s − 1.44e3·85-s − 320·89-s + 2.00e3·95-s + 248·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.20·11-s − 0.938·13-s − 1.02·17-s + 1.20·19-s + 1.08·23-s + 11/5·25-s − 1.39·29-s − 1.62·31-s − 0.133·37-s − 0.457·41-s + 0.780·43-s − 0.273·47-s − 0.285·53-s − 2.15·55-s − 1.27·59-s + 0.797·61-s − 1.67·65-s − 1.78·67-s + 0.187·71-s − 1.02·73-s − 0.694·79-s − 0.872·83-s − 1.83·85-s − 0.381·89-s + 2.15·95-s + 0.259·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 44 T + p^{3} T^{2} \) |
| 17 | \( 1 + 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 218 T + p^{3} T^{2} \) |
| 31 | \( 1 + 280 T + p^{3} T^{2} \) |
| 37 | \( 1 + 30 T + p^{3} T^{2} \) |
| 41 | \( 1 + 120 T + p^{3} T^{2} \) |
| 43 | \( 1 - 220 T + p^{3} T^{2} \) |
| 47 | \( 1 + 88 T + p^{3} T^{2} \) |
| 53 | \( 1 + 110 T + p^{3} T^{2} \) |
| 59 | \( 1 + 580 T + p^{3} T^{2} \) |
| 61 | \( 1 - 380 T + p^{3} T^{2} \) |
| 67 | \( 1 + 980 T + p^{3} T^{2} \) |
| 71 | \( 1 - 112 T + p^{3} T^{2} \) |
| 73 | \( 1 + 640 T + p^{3} T^{2} \) |
| 79 | \( 1 + 488 T + p^{3} T^{2} \) |
| 83 | \( 1 + 660 T + p^{3} T^{2} \) |
| 89 | \( 1 + 320 T + p^{3} T^{2} \) |
| 97 | \( 1 - 248 T + p^{3} T^{2} \) |
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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
−8.876773211852689029295496327903, −7.51931417997634608047466272975, −7.05536129368666403548666751415, −5.89614190493857664150055268964, −5.37665403179725781443554460367, −4.73468697094938213078097199805, −3.12685146282318003479667885586, −2.35795000589208637979510405165, −1.54423732694106313288872672972, 0,
1.54423732694106313288872672972, 2.35795000589208637979510405165, 3.12685146282318003479667885586, 4.73468697094938213078097199805, 5.37665403179725781443554460367, 5.89614190493857664150055268964, 7.05536129368666403548666751415, 7.51931417997634608047466272975, 8.876773211852689029295496327903
