| L(s) = 1 | − 1.56·3-s − 5-s + 3.56·7-s − 0.561·9-s + 11-s − 3.12·13-s + 1.56·15-s + 5.56·17-s + 2.43·19-s − 5.56·21-s + 7.12·23-s + 25-s + 5.56·27-s − 0.438·29-s + 8.68·31-s − 1.56·33-s − 3.56·35-s + 9.80·37-s + 4.87·39-s − 10·41-s + 5.12·43-s + 0.561·45-s − 7.12·47-s + 5.68·49-s − 8.68·51-s − 4.43·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.901·3-s − 0.447·5-s + 1.34·7-s − 0.187·9-s + 0.301·11-s − 0.866·13-s + 0.403·15-s + 1.34·17-s + 0.559·19-s − 1.21·21-s + 1.48·23-s + 0.200·25-s + 1.07·27-s − 0.0814·29-s + 1.55·31-s − 0.271·33-s − 0.602·35-s + 1.61·37-s + 0.780·39-s − 1.56·41-s + 0.781·43-s + 0.0837·45-s − 1.03·47-s + 0.812·49-s − 1.21·51-s − 0.609·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.089650462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.089650462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 + 0.438T + 29T^{2} \) |
| 31 | \( 1 - 8.68T + 31T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 - 0.876T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−11.32727325177861785739271547624, −10.47606385823248029619503964560, −9.390058851053015509560645800702, −8.175465431918091027579339278503, −7.55195342885962403559715939587, −6.36105976495826551067146152690, −5.16442569801710598650381886413, −4.70478840644818187394623215524, −3.02048209184950872865598493474, −1.10735599681963619788360581464,
1.10735599681963619788360581464, 3.02048209184950872865598493474, 4.70478840644818187394623215524, 5.16442569801710598650381886413, 6.36105976495826551067146152690, 7.55195342885962403559715939587, 8.175465431918091027579339278503, 9.390058851053015509560645800702, 10.47606385823248029619503964560, 11.32727325177861785739271547624
