| 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
−8.338408491410278861646422532835, −7.83012565926400065377838374311, −7.20784345520816286961368359840, −6.16463128814639741410425322258, −5.13937182507745925989313342217, −4.27454133299616004984499621811, −3.52643981168499783912225958304, −2.67054787423597735511395291669, −1.01083826114908052983380987936, 0,
1.01083826114908052983380987936, 2.67054787423597735511395291669, 3.52643981168499783912225958304, 4.27454133299616004984499621811, 5.13937182507745925989313342217, 6.16463128814639741410425322258, 7.20784345520816286961368359840, 7.83012565926400065377838374311, 8.338408491410278861646422532835
