| L(s) = 1 | + 1.94·3-s − 3.68·5-s − 2.51·7-s + 0.770·9-s + 2.53·11-s + 1.40·13-s − 7.15·15-s + 6.24·17-s + 6.03·19-s − 4.88·21-s − 0.990·23-s + 8.55·25-s − 4.32·27-s + 2.81·29-s + 8.19·31-s + 4.91·33-s + 9.26·35-s − 5.90·37-s + 2.72·39-s + 11.9·41-s − 7.71·43-s − 2.83·45-s − 11.4·47-s − 0.675·49-s + 12.1·51-s + 13.3·53-s − 9.32·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s − 1.64·5-s − 0.950·7-s + 0.256·9-s + 0.763·11-s + 0.388·13-s − 1.84·15-s + 1.51·17-s + 1.38·19-s − 1.06·21-s − 0.206·23-s + 1.71·25-s − 0.833·27-s + 0.522·29-s + 1.47·31-s + 0.855·33-s + 1.56·35-s − 0.971·37-s + 0.436·39-s + 1.85·41-s − 1.17·43-s − 0.422·45-s − 1.66·47-s − 0.0965·49-s + 1.69·51-s + 1.83·53-s − 1.25·55-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
| \(L(1)\) |
\(\approx\) |
\(1.736205508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736205508\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.94T + 3T^{2} \) |
| 5 | \( 1 + 3.68T + 5T^{2} \) |
| 7 | \( 1 + 2.51T + 7T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 - 6.03T + 19T^{2} \) |
| 23 | \( 1 + 0.990T + 23T^{2} \) |
| 29 | \( 1 - 2.81T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 + 5.90T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 7.71T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 5.22T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 8.02T + 79T^{2} \) |
| 83 | \( 1 + 8.79T + 83T^{2} \) |
| 89 | \( 1 + 8.44T + 89T^{2} \) |
| 97 | \( 1 - 3.29T + 97T^{2} \) |
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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
−9.605019673728936951204347517237, −8.560522381793952243659357320344, −8.148539315885339744179523757959, −7.36981842949359397039920861447, −6.61054942499210962070124017626, −5.36396717595794799157273539542, −3.99236380584660112494209816085, −3.48658528809858754299586765306, −2.87776047521485678437424041951, −0.936981695654590475031906063505,
0.936981695654590475031906063505, 2.87776047521485678437424041951, 3.48658528809858754299586765306, 3.99236380584660112494209816085, 5.36396717595794799157273539542, 6.61054942499210962070124017626, 7.36981842949359397039920861447, 8.148539315885339744179523757959, 8.560522381793952243659357320344, 9.605019673728936951204347517237
