| L(s) = 1 | + 5·5-s − 14·7-s + 6·11-s + 68·13-s − 78·17-s − 44·19-s + 120·23-s + 25·25-s − 126·29-s + 244·31-s − 70·35-s − 304·37-s + 480·41-s − 104·43-s + 600·47-s − 147·49-s + 258·53-s + 30·55-s + 534·59-s + 362·61-s + 340·65-s + 268·67-s − 972·71-s + 470·73-s − 84·77-s − 1.24e3·79-s + 396·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.164·11-s + 1.45·13-s − 1.11·17-s − 0.531·19-s + 1.08·23-s + 1/5·25-s − 0.806·29-s + 1.41·31-s − 0.338·35-s − 1.35·37-s + 1.82·41-s − 0.368·43-s + 1.86·47-s − 3/7·49-s + 0.668·53-s + 0.0735·55-s + 1.17·59-s + 0.759·61-s + 0.648·65-s + 0.488·67-s − 1.62·71-s + 0.753·73-s − 0.124·77-s − 1.77·79-s + 0.523·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.060417591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.060417591\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 68 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 126 T + p^{3} T^{2} \) |
| 31 | \( 1 - 244 T + p^{3} T^{2} \) |
| 37 | \( 1 + 304 T + p^{3} T^{2} \) |
| 41 | \( 1 - 480 T + p^{3} T^{2} \) |
| 43 | \( 1 + 104 T + p^{3} T^{2} \) |
| 47 | \( 1 - 600 T + p^{3} T^{2} \) |
| 53 | \( 1 - 258 T + p^{3} T^{2} \) |
| 59 | \( 1 - 534 T + p^{3} T^{2} \) |
| 61 | \( 1 - 362 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 71 | \( 1 + 972 T + p^{3} T^{2} \) |
| 73 | \( 1 - 470 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1244 T + p^{3} T^{2} \) |
| 83 | \( 1 - 396 T + p^{3} T^{2} \) |
| 89 | \( 1 - 972 T + p^{3} T^{2} \) |
| 97 | \( 1 + 46 T + p^{3} T^{2} \) |
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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
−10.05382074208939394635026393311, −8.964420673411454380233860357584, −8.655409251655525632856860891299, −7.20777975797714649638427767759, −6.42493483199127401119070966294, −5.74237118502087885495443180463, −4.43470751510138670131690785994, −3.43621740286061201520810999956, −2.24806435788978748436312181830, −0.832184243272800846755151944932,
0.832184243272800846755151944932, 2.24806435788978748436312181830, 3.43621740286061201520810999956, 4.43470751510138670131690785994, 5.74237118502087885495443180463, 6.42493483199127401119070966294, 7.20777975797714649638427767759, 8.655409251655525632856860891299, 8.964420673411454380233860357584, 10.05382074208939394635026393311
