L(s) = 1 | − 2.48·2-s + 1.56·3-s + 4.16·4-s + 3.76·5-s − 3.88·6-s − 3.67·7-s − 5.37·8-s − 0.552·9-s − 9.35·10-s + 5.01·11-s + 6.51·12-s + 1.93·13-s + 9.13·14-s + 5.89·15-s + 5.02·16-s − 0.320·17-s + 1.37·18-s − 2.85·19-s + 15.6·20-s − 5.75·21-s − 12.4·22-s − 1.64·23-s − 8.41·24-s + 9.18·25-s − 4.80·26-s − 5.55·27-s − 15.3·28-s + ⋯ |
L(s) = 1 | − 1.75·2-s + 0.903·3-s + 2.08·4-s + 1.68·5-s − 1.58·6-s − 1.39·7-s − 1.90·8-s − 0.184·9-s − 2.95·10-s + 1.51·11-s + 1.88·12-s + 0.536·13-s + 2.44·14-s + 1.52·15-s + 1.25·16-s − 0.0777·17-s + 0.323·18-s − 0.655·19-s + 3.50·20-s − 1.25·21-s − 2.65·22-s − 0.343·23-s − 1.71·24-s + 1.83·25-s − 0.942·26-s − 1.06·27-s − 2.89·28-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
\(L(1)\) |
\(\approx\) |
\(0.7800788364\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7800788364\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.48T + 2T^{2} \) |
| 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 - 3.76T + 5T^{2} \) |
| 7 | \( 1 + 3.67T + 7T^{2} \) |
| 11 | \( 1 - 5.01T + 11T^{2} \) |
| 13 | \( 1 - 1.93T + 13T^{2} \) |
| 17 | \( 1 + 0.320T + 17T^{2} \) |
| 19 | \( 1 + 2.85T + 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 - 8.72T + 29T^{2} \) |
| 31 | \( 1 - 1.09T + 31T^{2} \) |
| 37 | \( 1 + 4.88T + 37T^{2} \) |
| 41 | \( 1 + 9.59T + 41T^{2} \) |
| 43 | \( 1 + 5.85T + 43T^{2} \) |
| 47 | \( 1 + 6.33T + 47T^{2} \) |
| 53 | \( 1 - 6.55T + 53T^{2} \) |
| 59 | \( 1 + 6.51T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 0.221T + 67T^{2} \) |
| 71 | \( 1 + 4.41T + 71T^{2} \) |
| 73 | \( 1 + 1.92T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 5.44T + 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−13.38693930951906590344494156358, −11.96019203140438444434046166715, −10.42864224550182701318616246938, −9.769867156770577049998383858466, −9.067347062107784263557270771417, −8.517203032410962111112840874391, −6.65603768417448265693337110945, −6.27196928265547203409141944476, −3.10305270973767044169088182244, −1.75394998814133913255523014847,
1.75394998814133913255523014847, 3.10305270973767044169088182244, 6.27196928265547203409141944476, 6.65603768417448265693337110945, 8.517203032410962111112840874391, 9.067347062107784263557270771417, 9.769867156770577049998383858466, 10.42864224550182701318616246938, 11.96019203140438444434046166715, 13.38693930951906590344494156358