| L(s) = 1 | + 2-s − 2.56·3-s + 4-s − 1.19·5-s − 2.56·6-s + 8-s + 3.57·9-s − 1.19·10-s − 2.08·11-s − 2.56·12-s − 6.11·13-s + 3.05·15-s + 16-s + 3.68·17-s + 3.57·18-s − 2.27·19-s − 1.19·20-s − 2.08·22-s − 7.64·23-s − 2.56·24-s − 3.57·25-s − 6.11·26-s − 1.48·27-s + 2.05·29-s + 3.05·30-s − 31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.48·3-s + 0.5·4-s − 0.533·5-s − 1.04·6-s + 0.353·8-s + 1.19·9-s − 0.376·10-s − 0.628·11-s − 0.740·12-s − 1.69·13-s + 0.789·15-s + 0.250·16-s + 0.893·17-s + 0.843·18-s − 0.521·19-s − 0.266·20-s − 0.444·22-s − 1.59·23-s − 0.523·24-s − 0.715·25-s − 1.19·26-s − 0.285·27-s + 0.381·29-s + 0.558·30-s − 0.179·31-s + 0.176·32-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
| \(L(1)\) |
\(\approx\) |
\(0.7236589673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7236589673\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 1.19T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.08T + 11T^{2} \) |
| 13 | \( 1 + 6.11T + 13T^{2} \) |
| 17 | \( 1 - 3.68T + 17T^{2} \) |
| 19 | \( 1 + 2.27T + 19T^{2} \) |
| 23 | \( 1 + 7.64T + 23T^{2} \) |
| 29 | \( 1 - 2.05T + 29T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 2.08T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 9.01T + 59T^{2} \) |
| 61 | \( 1 - 3.03T + 61T^{2} \) |
| 67 | \( 1 + 9.34T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 2.02T + 79T^{2} \) |
| 83 | \( 1 - 6.95T + 83T^{2} \) |
| 89 | \( 1 - 5.81T + 89T^{2} \) |
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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
−7.69420953713518885812763232759, −7.29814279468966500154352342983, −6.51291536793494643247879343980, −5.65086640417123901609867159869, −5.38921296658026584608781823835, −4.53034583623858744489602534577, −3.99437184105129699067730598371, −2.84319259689133707543710568908, −1.89961695232508445556965562686, −0.41238170397150911278123351925,
0.41238170397150911278123351925, 1.89961695232508445556965562686, 2.84319259689133707543710568908, 3.99437184105129699067730598371, 4.53034583623858744489602534577, 5.38921296658026584608781823835, 5.65086640417123901609867159869, 6.51291536793494643247879343980, 7.29814279468966500154352342983, 7.69420953713518885812763232759
