| L(s) = 1 | − 1.68·2-s + 0.556·3-s + 0.843·4-s − 0.525·5-s − 0.937·6-s + 4.12·7-s + 1.95·8-s − 2.69·9-s + 0.885·10-s − 4.25·11-s + 0.469·12-s + 13-s − 6.95·14-s − 0.292·15-s − 4.97·16-s − 2.96·17-s + 4.53·18-s + 5.87·19-s − 0.442·20-s + 2.29·21-s + 7.17·22-s − 6.60·23-s + 1.08·24-s − 4.72·25-s − 1.68·26-s − 3.16·27-s + 3.47·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.321·3-s + 0.421·4-s − 0.234·5-s − 0.382·6-s + 1.55·7-s + 0.689·8-s − 0.896·9-s + 0.279·10-s − 1.28·11-s + 0.135·12-s + 0.277·13-s − 1.85·14-s − 0.0754·15-s − 1.24·16-s − 0.719·17-s + 1.06·18-s + 1.34·19-s − 0.0990·20-s + 0.500·21-s + 1.52·22-s − 1.37·23-s + 0.221·24-s − 0.944·25-s − 0.330·26-s − 0.609·27-s + 0.657·28-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 | \( 1 + 1.68T + 2T^{2} \) |
| 3 | \( 1 - 0.556T + 3T^{2} \) |
| 5 | \( 1 + 0.525T + 5T^{2} \) |
| 7 | \( 1 - 4.12T + 7T^{2} \) |
| 11 | \( 1 + 4.25T + 11T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 + 6.60T + 23T^{2} \) |
| 29 | \( 1 + 0.311T + 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 - 3.25T + 37T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 + 0.111T + 43T^{2} \) |
| 47 | \( 1 - 4.07T + 47T^{2} \) |
| 53 | \( 1 - 2.52T + 53T^{2} \) |
| 59 | \( 1 - 2.38T + 59T^{2} \) |
| 61 | \( 1 - 6.57T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 6.12T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 3.86T + 79T^{2} \) |
| 83 | \( 1 + 8.57T + 83T^{2} \) |
| 89 | \( 1 - 0.219T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−8.953620578472793290102760863883, −8.411669184043734510435028113951, −7.78629827495234687470847818317, −7.43229709634215559867851407048, −5.79674139772636023885930610803, −5.04396023701398163676069168162, −4.04210528249504221256478667289, −2.56169955492559586666026213577, −1.60455200530127341747975663561, 0,
1.60455200530127341747975663561, 2.56169955492559586666026213577, 4.04210528249504221256478667289, 5.04396023701398163676069168162, 5.79674139772636023885930610803, 7.43229709634215559867851407048, 7.78629827495234687470847818317, 8.411669184043734510435028113951, 8.953620578472793290102760863883
