| L(s) = 1 | − 2-s + 1.43·3-s + 4-s − 1.66·5-s − 1.43·6-s + 4.92·7-s − 8-s − 0.954·9-s + 1.66·10-s − 4.36·11-s + 1.43·12-s + 0.747·13-s − 4.92·14-s − 2.38·15-s + 16-s + 3.71·17-s + 0.954·18-s + 8.35·19-s − 1.66·20-s + 7.04·21-s + 4.36·22-s + 5.36·23-s − 1.43·24-s − 2.22·25-s − 0.747·26-s − 5.65·27-s + 4.92·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.825·3-s + 0.5·4-s − 0.745·5-s − 0.583·6-s + 1.86·7-s − 0.353·8-s − 0.318·9-s + 0.527·10-s − 1.31·11-s + 0.412·12-s + 0.207·13-s − 1.31·14-s − 0.615·15-s + 0.250·16-s + 0.901·17-s + 0.224·18-s + 1.91·19-s − 0.372·20-s + 1.53·21-s + 0.931·22-s + 1.11·23-s − 0.291·24-s − 0.444·25-s − 0.146·26-s − 1.08·27-s + 0.930·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.037502824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.037502824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 1.43T + 3T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 13 | \( 1 - 0.747T + 13T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 - 8.35T + 19T^{2} \) |
| 23 | \( 1 - 5.36T + 23T^{2} \) |
| 29 | \( 1 - 0.205T + 29T^{2} \) |
| 37 | \( 1 - 9.33T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 + 7.07T + 43T^{2} \) |
| 47 | \( 1 - 0.917T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 7.68T + 59T^{2} \) |
| 61 | \( 1 - 8.14T + 61T^{2} \) |
| 67 | \( 1 - 3.53T + 67T^{2} \) |
| 71 | \( 1 - 6.95T + 71T^{2} \) |
| 73 | \( 1 + 6.79T + 73T^{2} \) |
| 79 | \( 1 + 9.04T + 79T^{2} \) |
| 83 | \( 1 + 9.70T + 83T^{2} \) |
| 89 | \( 1 + 0.507T + 89T^{2} \) |
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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
−8.092015168377123844280407085274, −7.61441357919889525906650176350, −7.32879308042270503386761100321, −5.75466899580853387408828987318, −5.25811592230847847402499668795, −4.46984452486456075726112252892, −3.31457614011053964808118833350, −2.80310942213263772184481363726, −1.77311314822365748922954340294, −0.831193944715262212199478620582,
0.831193944715262212199478620582, 1.77311314822365748922954340294, 2.80310942213263772184481363726, 3.31457614011053964808118833350, 4.46984452486456075726112252892, 5.25811592230847847402499668795, 5.75466899580853387408828987318, 7.32879308042270503386761100321, 7.61441357919889525906650176350, 8.092015168377123844280407085274
