L(s) = 1 | + 2.10·2-s − 2.04·3-s + 2.42·4-s + 2.55·5-s − 4.30·6-s − 3.75·7-s + 0.901·8-s + 1.18·9-s + 5.36·10-s − 4.96·12-s − 4.57·13-s − 7.90·14-s − 5.22·15-s − 2.95·16-s − 3.31·17-s + 2.49·18-s − 4.20·19-s + 6.19·20-s + 7.68·21-s + 6.43·23-s − 1.84·24-s + 1.50·25-s − 9.62·26-s + 3.70·27-s − 9.12·28-s + 3.81·29-s − 10.9·30-s + ⋯ |
L(s) = 1 | + 1.48·2-s − 1.18·3-s + 1.21·4-s + 1.14·5-s − 1.75·6-s − 1.42·7-s + 0.318·8-s + 0.395·9-s + 1.69·10-s − 1.43·12-s − 1.26·13-s − 2.11·14-s − 1.34·15-s − 0.739·16-s − 0.804·17-s + 0.589·18-s − 0.964·19-s + 1.38·20-s + 1.67·21-s + 1.34·23-s − 0.376·24-s + 0.301·25-s − 1.88·26-s + 0.713·27-s − 1.72·28-s + 0.708·29-s − 2.00·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.071609763\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.071609763\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 2.10T + 2T^{2} \) |
| 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 - 2.55T + 5T^{2} \) |
| 7 | \( 1 + 3.75T + 7T^{2} \) |
| 13 | \( 1 + 4.57T + 13T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 + 4.20T + 19T^{2} \) |
| 23 | \( 1 - 6.43T + 23T^{2} \) |
| 29 | \( 1 - 3.81T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 - 6.38T + 41T^{2} \) |
| 43 | \( 1 - 8.90T + 43T^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 - 1.65T + 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 5.86T + 71T^{2} \) |
| 73 | \( 1 + 5.92T + 73T^{2} \) |
| 79 | \( 1 - 5.87T + 79T^{2} \) |
| 83 | \( 1 - 7.55T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 3.56T + 97T^{2} \) |
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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
−7.29411418005596699139539868941, −6.73202286790933314722951662773, −6.20455943619879382725567321512, −5.86229263956202217258892228710, −5.10325532393349413578115255763, −4.63029729831904070077032486135, −3.72165514048843387749526309665, −2.63604599751143702766004205614, −2.34588076108262642903505109076, −0.57103748522203767873027336397,
0.57103748522203767873027336397, 2.34588076108262642903505109076, 2.63604599751143702766004205614, 3.72165514048843387749526309665, 4.63029729831904070077032486135, 5.10325532393349413578115255763, 5.86229263956202217258892228710, 6.20455943619879382725567321512, 6.73202286790933314722951662773, 7.29411418005596699139539868941