| L(s) = 1 | + 2.28·2-s − 3.03·3-s + 3.22·4-s + 3.97·5-s − 6.94·6-s − 0.589·7-s + 2.79·8-s + 6.21·9-s + 9.08·10-s − 1.38·11-s − 9.79·12-s − 3.36·13-s − 1.34·14-s − 12.0·15-s − 0.0515·16-s − 7.97·17-s + 14.2·18-s + 1.27·19-s + 12.8·20-s + 1.78·21-s − 3.17·22-s + 0.987·23-s − 8.49·24-s + 10.8·25-s − 7.69·26-s − 9.77·27-s − 1.90·28-s + ⋯ |
| L(s) = 1 | + 1.61·2-s − 1.75·3-s + 1.61·4-s + 1.77·5-s − 2.83·6-s − 0.222·7-s + 0.989·8-s + 2.07·9-s + 2.87·10-s − 0.418·11-s − 2.82·12-s − 0.933·13-s − 0.359·14-s − 3.11·15-s − 0.0128·16-s − 1.93·17-s + 3.35·18-s + 0.292·19-s + 2.86·20-s + 0.390·21-s − 0.677·22-s + 0.205·23-s − 1.73·24-s + 2.16·25-s − 1.50·26-s − 1.88·27-s − 0.359·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780806073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780806073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 139 | \( 1 - T \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 + 3.03T + 3T^{2} \) |
| 5 | \( 1 - 3.97T + 5T^{2} \) |
| 7 | \( 1 + 0.589T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 + 7.97T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 0.987T + 23T^{2} \) |
| 29 | \( 1 - 3.80T + 29T^{2} \) |
| 31 | \( 1 - 2.49T + 31T^{2} \) |
| 37 | \( 1 - 4.32T + 37T^{2} \) |
| 41 | \( 1 - 0.260T + 41T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 - 8.41T + 53T^{2} \) |
| 59 | \( 1 - 4.54T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 7.40T + 67T^{2} \) |
| 71 | \( 1 - 6.43T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 6.81T + 79T^{2} \) |
| 83 | \( 1 - 8.02T + 83T^{2} \) |
| 89 | \( 1 - 2.71T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−13.12870505338129325796123193347, −12.44367113862072751193126124688, −11.38164927016222212567588468595, −10.51987648643328951662529690708, −9.521374341158961387754949244192, −6.72640172004130596404443298421, −6.32942821830898514877625774279, −5.25608842699554926932562055754, −4.71885933750437545998620785463, −2.33264701009223557130826448329,
2.33264701009223557130826448329, 4.71885933750437545998620785463, 5.25608842699554926932562055754, 6.32942821830898514877625774279, 6.72640172004130596404443298421, 9.521374341158961387754949244192, 10.51987648643328951662529690708, 11.38164927016222212567588468595, 12.44367113862072751193126124688, 13.12870505338129325796123193347
