| L(s) = 1 | − 1.56·3-s + 5-s − 1.56·7-s − 0.561·9-s − 4·11-s − 6.68·13-s − 1.56·15-s + 7.56·17-s + 19-s + 2.43·21-s + 4.68·23-s + 25-s + 5.56·27-s + 6.68·29-s − 3.12·31-s + 6.24·33-s − 1.56·35-s − 6·37-s + 10.4·39-s − 4.24·41-s + 11.1·43-s − 0.561·45-s + 10.2·47-s − 4.56·49-s − 11.8·51-s − 0.438·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.447·5-s − 0.590·7-s − 0.187·9-s − 1.20·11-s − 1.85·13-s − 0.403·15-s + 1.83·17-s + 0.229·19-s + 0.532·21-s + 0.976·23-s + 0.200·25-s + 1.07·27-s + 1.24·29-s − 0.560·31-s + 1.08·33-s − 0.263·35-s − 0.986·37-s + 1.67·39-s − 0.663·41-s + 1.69·43-s − 0.0837·45-s + 1.49·47-s − 0.651·49-s − 1.65·51-s − 0.0602·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8886074347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8886074347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 6.68T + 13T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 0.438T + 53T^{2} \) |
| 59 | \( 1 - 1.56T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 + 1.56T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−9.782058450338128769646895071040, −8.742833855301265300468674316697, −7.63080759639164798043466291501, −7.08932495375846240125772557179, −6.01406856163716266316444291209, −5.28866707645966857207009179139, −4.90996661382108893231912606206, −3.21542525101062319796510249844, −2.48182177740369276017061150106, −0.67172959946860338931861754279,
0.67172959946860338931861754279, 2.48182177740369276017061150106, 3.21542525101062319796510249844, 4.90996661382108893231912606206, 5.28866707645966857207009179139, 6.01406856163716266316444291209, 7.08932495375846240125772557179, 7.63080759639164798043466291501, 8.742833855301265300468674316697, 9.782058450338128769646895071040
