| L(s) = 1 | − 2.31·3-s − 1.78·5-s − 4.41·7-s + 2.35·9-s − 3.00·11-s − 0.0205·13-s + 4.13·15-s − 4.88·17-s − 5.01·19-s + 10.2·21-s − 3.08·23-s − 1.80·25-s + 1.48·27-s + 3.93·29-s − 4.09·31-s + 6.95·33-s + 7.89·35-s − 3.21·37-s + 0.0475·39-s + 3.58·41-s + 5.15·43-s − 4.21·45-s − 10.0·47-s + 12.5·49-s + 11.3·51-s + 0.412·53-s + 5.36·55-s + ⋯ |
| L(s) = 1 | − 1.33·3-s − 0.798·5-s − 1.67·7-s + 0.786·9-s − 0.906·11-s − 0.00569·13-s + 1.06·15-s − 1.18·17-s − 1.15·19-s + 2.23·21-s − 0.643·23-s − 0.361·25-s + 0.285·27-s + 0.731·29-s − 0.734·31-s + 1.21·33-s + 1.33·35-s − 0.528·37-s + 0.00761·39-s + 0.559·41-s + 0.786·43-s − 0.627·45-s − 1.46·47-s + 1.79·49-s + 1.58·51-s + 0.0566·53-s + 0.724·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1240687566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1240687566\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 3 | \( 1 + 2.31T + 3T^{2} \) |
| 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 13 | \( 1 + 0.0205T + 13T^{2} \) |
| 17 | \( 1 + 4.88T + 17T^{2} \) |
| 19 | \( 1 + 5.01T + 19T^{2} \) |
| 23 | \( 1 + 3.08T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 + 3.21T + 37T^{2} \) |
| 41 | \( 1 - 3.58T + 41T^{2} \) |
| 43 | \( 1 - 5.15T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 0.412T + 53T^{2} \) |
| 59 | \( 1 + 2.41T + 59T^{2} \) |
| 61 | \( 1 - 9.25T + 61T^{2} \) |
| 67 | \( 1 + 5.90T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 + 7.24T + 73T^{2} \) |
| 79 | \( 1 - 4.80T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 7.38T + 89T^{2} \) |
| 97 | \( 1 + 5.55T + 97T^{2} \) |
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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
−9.840697986805508699118930930638, −8.850134919411601184312331218059, −7.906947297250347867820322361106, −6.82585834838098014150386736594, −6.39598148303554732489952838279, −5.58164933058979750539759297032, −4.54199336811151748940938918232, −3.68415150856816231543125342833, −2.46706728195288175370077846436, −0.25258999586443835393340449380,
0.25258999586443835393340449380, 2.46706728195288175370077846436, 3.68415150856816231543125342833, 4.54199336811151748940938918232, 5.58164933058979750539759297032, 6.39598148303554732489952838279, 6.82585834838098014150386736594, 7.906947297250347867820322361106, 8.850134919411601184312331218059, 9.840697986805508699118930930638
