L(s) = 1 | − 1.68·2-s − 3-s + 0.822·4-s + 0.771·5-s + 1.68·6-s + 7-s + 1.97·8-s + 9-s − 1.29·10-s − 4.02·11-s − 0.822·12-s + 13-s − 1.68·14-s − 0.771·15-s − 4.96·16-s + 17-s − 1.68·18-s − 7.14·19-s + 0.634·20-s − 21-s + 6.77·22-s − 0.861·23-s − 1.97·24-s − 4.40·25-s − 1.68·26-s − 27-s + 0.822·28-s + ⋯ |
L(s) = 1 | − 1.18·2-s − 0.577·3-s + 0.411·4-s + 0.345·5-s + 0.685·6-s + 0.377·7-s + 0.699·8-s + 0.333·9-s − 0.409·10-s − 1.21·11-s − 0.237·12-s + 0.277·13-s − 0.449·14-s − 0.199·15-s − 1.24·16-s + 0.242·17-s − 0.395·18-s − 1.64·19-s + 0.141·20-s − 0.218·21-s + 1.44·22-s − 0.179·23-s − 0.403·24-s − 0.880·25-s − 0.329·26-s − 0.192·27-s + 0.155·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4980547810\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4980547810\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + 1.68T + 2T^{2} \) |
| 5 | \( 1 - 0.771T + 5T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 19 | \( 1 + 7.14T + 19T^{2} \) |
| 23 | \( 1 + 0.861T + 23T^{2} \) |
| 29 | \( 1 + 9.84T + 29T^{2} \) |
| 31 | \( 1 - 0.961T + 31T^{2} \) |
| 37 | \( 1 - 0.267T + 37T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 - 6.27T + 47T^{2} \) |
| 53 | \( 1 + 0.916T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 5.22T + 61T^{2} \) |
| 67 | \( 1 - 7.95T + 67T^{2} \) |
| 71 | \( 1 + 3.31T + 71T^{2} \) |
| 73 | \( 1 - 9.51T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.171677273778772336141937792209, −7.82325322713673220265749768634, −7.12985550735415611825532756919, −6.09073658276960441828518658546, −5.57068531656635894993419243982, −4.62806488497197833689056500156, −3.93671639238360790865486842233, −2.39958866834725982225845143352, −1.74537363158730391268747747041, −0.47958243441501779496469187874,
0.47958243441501779496469187874, 1.74537363158730391268747747041, 2.39958866834725982225845143352, 3.93671639238360790865486842233, 4.62806488497197833689056500156, 5.57068531656635894993419243982, 6.09073658276960441828518658546, 7.12985550735415611825532756919, 7.82325322713673220265749768634, 8.171677273778772336141937792209