L(s) = 1 | + 2-s + 2.11·3-s + 4-s − 1.27·5-s + 2.11·6-s + 2.13·7-s + 8-s + 1.47·9-s − 1.27·10-s + 4.19·11-s + 2.11·12-s − 4.74·13-s + 2.13·14-s − 2.68·15-s + 16-s + 0.786·17-s + 1.47·18-s + 2.45·19-s − 1.27·20-s + 4.52·21-s + 4.19·22-s + 3.93·23-s + 2.11·24-s − 3.38·25-s − 4.74·26-s − 3.23·27-s + 2.13·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.22·3-s + 0.5·4-s − 0.568·5-s + 0.863·6-s + 0.808·7-s + 0.353·8-s + 0.490·9-s − 0.401·10-s + 1.26·11-s + 0.610·12-s − 1.31·13-s + 0.571·14-s − 0.694·15-s + 0.250·16-s + 0.190·17-s + 0.347·18-s + 0.562·19-s − 0.284·20-s + 0.987·21-s + 0.895·22-s + 0.820·23-s + 0.431·24-s − 0.676·25-s − 0.930·26-s − 0.621·27-s + 0.404·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.039774246\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.039774246\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 - 2.11T + 3T^{2} \) |
| 5 | \( 1 + 1.27T + 5T^{2} \) |
| 7 | \( 1 - 2.13T + 7T^{2} \) |
| 11 | \( 1 - 4.19T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 - 0.786T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 23 | \( 1 - 3.93T + 23T^{2} \) |
| 29 | \( 1 - 9.32T + 29T^{2} \) |
| 31 | \( 1 - 8.02T + 31T^{2} \) |
| 37 | \( 1 - 2.00T + 37T^{2} \) |
| 41 | \( 1 - 3.68T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 + 9.05T + 47T^{2} \) |
| 53 | \( 1 + 1.37T + 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 - 0.763T + 61T^{2} \) |
| 67 | \( 1 + 5.66T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 7.89T + 97T^{2} \) |
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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
−9.453726086115504120972221451990, −8.459684618749338748008622571435, −7.945164176944110386982588506499, −7.18674521786305627550300221548, −6.32014672364769784604355003021, −4.94591620806436097212886589273, −4.41658864437689790155316713338, −3.37257817210999427297835143321, −2.65759858037867923751749824989, −1.44268565818194300487836720651,
1.44268565818194300487836720651, 2.65759858037867923751749824989, 3.37257817210999427297835143321, 4.41658864437689790155316713338, 4.94591620806436097212886589273, 6.32014672364769784604355003021, 7.18674521786305627550300221548, 7.945164176944110386982588506499, 8.459684618749338748008622571435, 9.453726086115504120972221451990