| L(s) = 1 | + 1.63·2-s + 2.37·3-s + 0.668·4-s − 3.50·5-s + 3.88·6-s − 4.23·7-s − 2.17·8-s + 2.65·9-s − 5.73·10-s + 2.48·11-s + 1.58·12-s + 13-s − 6.91·14-s − 8.33·15-s − 4.89·16-s − 6.44·17-s + 4.33·18-s + 1.06·19-s − 2.34·20-s − 10.0·21-s + 4.06·22-s + 0.0498·23-s − 5.16·24-s + 7.30·25-s + 1.63·26-s − 0.829·27-s − 2.82·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 1.37·3-s + 0.334·4-s − 1.56·5-s + 1.58·6-s − 1.59·7-s − 0.768·8-s + 0.883·9-s − 1.81·10-s + 0.750·11-s + 0.458·12-s + 0.277·13-s − 1.84·14-s − 2.15·15-s − 1.22·16-s − 1.56·17-s + 1.02·18-s + 0.245·19-s − 0.524·20-s − 2.19·21-s + 0.867·22-s + 0.0104·23-s − 1.05·24-s + 1.46·25-s + 0.320·26-s − 0.159·27-s − 0.534·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 3 | \( 1 - 2.37T + 3T^{2} \) |
| 5 | \( 1 + 3.50T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 - 0.0498T + 23T^{2} \) |
| 29 | \( 1 + 6.02T + 29T^{2} \) |
| 31 | \( 1 + 3.20T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 - 2.21T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 8.62T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 8.60T + 61T^{2} \) |
| 67 | \( 1 - 2.13T + 67T^{2} \) |
| 71 | \( 1 - 8.84T + 71T^{2} \) |
| 73 | \( 1 + 9.13T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.80T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 7.89T + 97T^{2} \) |
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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
−8.976548645497084786482143480554, −8.642867040113060842503867331710, −7.42195565388192510095558120518, −6.82969110050976347555874451608, −5.88717414155496075145971104003, −4.41432874279471818408509237280, −3.77847368720495046368320890248, −3.42953382975674660970004987594, −2.50045573945173515430427492370, 0,
2.50045573945173515430427492370, 3.42953382975674660970004987594, 3.77847368720495046368320890248, 4.41432874279471818408509237280, 5.88717414155496075145971104003, 6.82969110050976347555874451608, 7.42195565388192510095558120518, 8.642867040113060842503867331710, 8.976548645497084786482143480554
