| L(s) = 1 | − 2.21·5-s − 4.42·7-s − 0.903·11-s + 0.622·13-s − 4.21·17-s − 5.80·19-s + 1.24·23-s − 0.0967·25-s − 8.85·29-s − 3.05·31-s + 9.80·35-s − 1.18·37-s − 2.02·41-s − 8.83·43-s − 3.09·47-s + 12.6·49-s + 4.34·53-s + 2·55-s + 12.4·59-s − 2.90·61-s − 1.37·65-s + 5.59·67-s + 4·71-s − 7.80·73-s + 4·77-s + 1.72·79-s − 7.47·83-s + ⋯ |
| L(s) = 1 | − 0.990·5-s − 1.67·7-s − 0.272·11-s + 0.172·13-s − 1.02·17-s − 1.33·19-s + 0.259·23-s − 0.0193·25-s − 1.64·29-s − 0.547·31-s + 1.65·35-s − 0.194·37-s − 0.315·41-s − 1.34·43-s − 0.451·47-s + 1.80·49-s + 0.597·53-s + 0.269·55-s + 1.61·59-s − 0.371·61-s − 0.170·65-s + 0.683·67-s + 0.474·71-s − 0.913·73-s + 0.455·77-s + 0.194·79-s − 0.820·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2142826121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2142826121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 + 0.903T + 11T^{2} \) |
| 13 | \( 1 - 0.622T + 13T^{2} \) |
| 17 | \( 1 + 4.21T + 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 + 1.18T + 37T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 + 8.83T + 43T^{2} \) |
| 47 | \( 1 + 3.09T + 47T^{2} \) |
| 53 | \( 1 - 4.34T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 2.90T + 61T^{2} \) |
| 67 | \( 1 - 5.59T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 - 1.72T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 + 7.28T + 97T^{2} \) |
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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
−8.175797488033433909104912530868, −7.14776912905252219079173761069, −6.82169452793044580027582647314, −6.07209263457006207690367153371, −5.25107627088333417742413073180, −4.11201258441599326333491277896, −3.77354136993346645666045609266, −2.92676938673234464202461887227, −1.96938113176195731561375231527, −0.22611135603423046277344123224,
0.22611135603423046277344123224, 1.96938113176195731561375231527, 2.92676938673234464202461887227, 3.77354136993346645666045609266, 4.11201258441599326333491277896, 5.25107627088333417742413073180, 6.07209263457006207690367153371, 6.82169452793044580027582647314, 7.14776912905252219079173761069, 8.175797488033433909104912530868
