L(s) = 1 | + 1.56·2-s + 3·3-s − 5.56·4-s + 4.68·6-s − 24.2·7-s − 21.1·8-s + 9·9-s − 11·11-s − 16.6·12-s − 84.2·13-s − 37.8·14-s + 11.4·16-s + 40.9·17-s + 14.0·18-s + 120.·19-s − 72.8·21-s − 17.1·22-s + 9.94·23-s − 63.5·24-s − 131.·26-s + 27·27-s + 134.·28-s + 196.·29-s + 151.·31-s + 187.·32-s − 33·33-s + 63.9·34-s + ⋯ |
L(s) = 1 | + 0.552·2-s + 0.577·3-s − 0.695·4-s + 0.318·6-s − 1.31·7-s − 0.935·8-s + 0.333·9-s − 0.301·11-s − 0.401·12-s − 1.79·13-s − 0.723·14-s + 0.178·16-s + 0.584·17-s + 0.184·18-s + 1.45·19-s − 0.756·21-s − 0.166·22-s + 0.0901·23-s − 0.540·24-s − 0.992·26-s + 0.192·27-s + 0.910·28-s + 1.26·29-s + 0.875·31-s + 1.03·32-s − 0.174·33-s + 0.322·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.784778172\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.784778172\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 - 1.56T + 8T^{2} \) |
| 7 | \( 1 + 24.2T + 343T^{2} \) |
| 13 | \( 1 + 84.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.94T + 1.21e4T^{2} \) |
| 29 | \( 1 - 196.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 253.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 90.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 491.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 127.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 628.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 87.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 390.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 165.T + 9.12e5T^{2} \) |
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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
−9.670739804650079207127555928671, −9.303668935663211238233415384912, −8.088086916422495041345867729408, −7.30345284274653734801275272324, −6.26115915164428246749622942199, −5.22973139072209495777267003216, −4.43208926252984556757243813048, −3.19132984510003715589776915432, −2.75482100987472045760953286303, −0.64765716805417955582301326791,
0.64765716805417955582301326791, 2.75482100987472045760953286303, 3.19132984510003715589776915432, 4.43208926252984556757243813048, 5.22973139072209495777267003216, 6.26115915164428246749622942199, 7.30345284274653734801275272324, 8.088086916422495041345867729408, 9.303668935663211238233415384912, 9.670739804650079207127555928671