| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s − 4·11-s − 2·15-s + 8·19-s + 2·21-s + 6·23-s − 25-s − 27-s + 10·29-s + 2·31-s + 4·33-s − 4·35-s + 6·37-s + 6·41-s − 12·43-s + 2·45-s + 4·47-s − 3·49-s − 4·53-s − 8·55-s − 8·57-s + 14·61-s − 2·63-s + 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.516·15-s + 1.83·19-s + 0.436·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s + 1.85·29-s + 0.359·31-s + 0.696·33-s − 0.676·35-s + 0.986·37-s + 0.937·41-s − 1.82·43-s + 0.298·45-s + 0.583·47-s − 3/7·49-s − 0.549·53-s − 1.07·55-s − 1.05·57-s + 1.79·61-s − 0.251·63-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.12358043453516, −12.80494237511053, −12.41812140179758, −11.60581402806995, −11.48921232429410, −10.83695122840328, −10.20327717568087, −9.971954505525381, −9.671413526011552, −9.139165220684915, −8.459587540374963, −7.982138808203020, −7.403099475892684, −6.931755530494956, −6.419601555051719, −6.004738958105661, −5.448112618335461, −4.992243049349001, −4.764203949685160, −3.777986059403075, −3.187936732916810, −2.693056309100028, −2.296898816044761, −1.216723928289328, −0.9236881777875847, 0,
0.9236881777875847, 1.216723928289328, 2.296898816044761, 2.693056309100028, 3.187936732916810, 3.777986059403075, 4.764203949685160, 4.992243049349001, 5.448112618335461, 6.004738958105661, 6.419601555051719, 6.931755530494956, 7.403099475892684, 7.982138808203020, 8.459587540374963, 9.139165220684915, 9.671413526011552, 9.971954505525381, 10.20327717568087, 10.83695122840328, 11.48921232429410, 11.60581402806995, 12.41812140179758, 12.80494237511053, 13.12358043453516
