| L(s) = 1 | + 2-s + 3.30·3-s + 4-s − 0.828·5-s + 3.30·6-s + 2.22·7-s + 8-s + 7.94·9-s − 0.828·10-s + 3.30·12-s + 1.20·13-s + 2.22·14-s − 2.74·15-s + 16-s + 4.58·17-s + 7.94·18-s + 19-s − 0.828·20-s + 7.37·21-s − 1.76·23-s + 3.30·24-s − 4.31·25-s + 1.20·26-s + 16.3·27-s + 2.22·28-s + 3.91·29-s − 2.74·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.90·3-s + 0.5·4-s − 0.370·5-s + 1.35·6-s + 0.842·7-s + 0.353·8-s + 2.64·9-s − 0.262·10-s + 0.954·12-s + 0.334·13-s + 0.595·14-s − 0.708·15-s + 0.250·16-s + 1.11·17-s + 1.87·18-s + 0.229·19-s − 0.185·20-s + 1.60·21-s − 0.367·23-s + 0.675·24-s − 0.862·25-s + 0.236·26-s + 3.14·27-s + 0.421·28-s + 0.726·29-s − 0.500·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.989656525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.989656525\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 + 0.828T + 5T^{2} \) |
| 7 | \( 1 - 2.22T + 7T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 - 4.58T + 17T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 + 5.91T + 41T^{2} \) |
| 43 | \( 1 + 3.78T + 43T^{2} \) |
| 47 | \( 1 + 7.92T + 47T^{2} \) |
| 53 | \( 1 + 8.13T + 53T^{2} \) |
| 59 | \( 1 + 0.104T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 5.40T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 2.32T + 79T^{2} \) |
| 83 | \( 1 + 5.98T + 83T^{2} \) |
| 89 | \( 1 + 6.07T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.352368940495281437590041861298, −7.56921489979576469632239701704, −7.27888924177797901602037475606, −6.15189308037131286122761251495, −5.08305035868433615721715993291, −4.40173039388513075336905263730, −3.50167336666125400965455626719, −3.24268466674604837641267754201, −2.03329567355089532686557666915, −1.47129154134962916779121015573,
1.47129154134962916779121015573, 2.03329567355089532686557666915, 3.24268466674604837641267754201, 3.50167336666125400965455626719, 4.40173039388513075336905263730, 5.08305035868433615721715993291, 6.15189308037131286122761251495, 7.27888924177797901602037475606, 7.56921489979576469632239701704, 8.352368940495281437590041861298
