| L(s) = 1 | − 7·3-s + 6·7-s + 22·9-s + 43·11-s + 28·13-s + 91·17-s + 35·19-s − 42·21-s + 162·23-s + 35·27-s − 160·29-s + 42·31-s − 301·33-s + 314·37-s − 196·39-s − 203·41-s − 92·43-s + 196·47-s − 307·49-s − 637·51-s − 82·53-s − 245·57-s + 280·59-s + 518·61-s + 132·63-s − 141·67-s − 1.13e3·69-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.323·7-s + 0.814·9-s + 1.17·11-s + 0.597·13-s + 1.29·17-s + 0.422·19-s − 0.436·21-s + 1.46·23-s + 0.249·27-s − 1.02·29-s + 0.243·31-s − 1.58·33-s + 1.39·37-s − 0.804·39-s − 0.773·41-s − 0.326·43-s + 0.608·47-s − 0.895·49-s − 1.74·51-s − 0.212·53-s − 0.569·57-s + 0.617·59-s + 1.08·61-s + 0.263·63-s − 0.257·67-s − 1.97·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.729367092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.729367092\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 43 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 91 T + p^{3} T^{2} \) |
| 19 | \( 1 - 35 T + p^{3} T^{2} \) |
| 23 | \( 1 - 162 T + p^{3} T^{2} \) |
| 29 | \( 1 + 160 T + p^{3} T^{2} \) |
| 31 | \( 1 - 42 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 203 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 - 196 T + p^{3} T^{2} \) |
| 53 | \( 1 + 82 T + p^{3} T^{2} \) |
| 59 | \( 1 - 280 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 141 T + p^{3} T^{2} \) |
| 71 | \( 1 - 412 T + p^{3} T^{2} \) |
| 73 | \( 1 + 763 T + p^{3} T^{2} \) |
| 79 | \( 1 - 510 T + p^{3} T^{2} \) |
| 83 | \( 1 + 777 T + p^{3} T^{2} \) |
| 89 | \( 1 + 945 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1246 T + p^{3} T^{2} \) |
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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
−9.162807130780010267449038509392, −8.220536188033643559884166462784, −7.22399100219122118214443336759, −6.52221224936827348001238382577, −5.72654269895957227979150128556, −5.13122971338231583607225141514, −4.13721208209058985945257453930, −3.14836832984906747400199666811, −1.44017897451635070895122931497, −0.77454565099657160217588920506,
0.77454565099657160217588920506, 1.44017897451635070895122931497, 3.14836832984906747400199666811, 4.13721208209058985945257453930, 5.13122971338231583607225141514, 5.72654269895957227979150128556, 6.52221224936827348001238382577, 7.22399100219122118214443336759, 8.220536188033643559884166462784, 9.162807130780010267449038509392
