L(s) = 1 | + 2.94·2-s − 3.08·3-s + 0.644·4-s + 5·5-s − 9.08·6-s + 21.8·7-s − 21.6·8-s − 17.4·9-s + 14.7·10-s + 11·11-s − 1.99·12-s + 82.8·13-s + 64.1·14-s − 15.4·15-s − 68.7·16-s + 12.6·17-s − 51.3·18-s − 19·19-s + 3.22·20-s − 67.4·21-s + 32.3·22-s − 184.·23-s + 66.8·24-s + 25·25-s + 243.·26-s + 137.·27-s + 14.0·28-s + ⋯ |
L(s) = 1 | + 1.03·2-s − 0.594·3-s + 0.0805·4-s + 0.447·5-s − 0.618·6-s + 1.17·7-s − 0.955·8-s − 0.646·9-s + 0.464·10-s + 0.301·11-s − 0.0479·12-s + 1.76·13-s + 1.22·14-s − 0.265·15-s − 1.07·16-s + 0.180·17-s − 0.671·18-s − 0.229·19-s + 0.0360·20-s − 0.700·21-s + 0.313·22-s − 1.66·23-s + 0.568·24-s + 0.200·25-s + 1.83·26-s + 0.979·27-s + 0.0949·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.186812219\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.186812219\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
good | 2 | \( 1 - 2.94T + 8T^{2} \) |
| 3 | \( 1 + 3.08T + 27T^{2} \) |
| 7 | \( 1 - 21.8T + 343T^{2} \) |
| 13 | \( 1 - 82.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.6T + 4.91e3T^{2} \) |
| 23 | \( 1 + 184.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 107.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 57.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 146.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 304.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 480.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 581.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 420.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 357.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 584.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.17e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 333.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 786.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 87.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.45e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3T + 9.12e5T^{2} \) |
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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
−9.496431189611638169176227558437, −8.593593995515546013560150047702, −8.040249755561486568046256570245, −6.42642510005325526840501326766, −5.97263589617596147119559165159, −5.27436486399365145320812776399, −4.36911497516141879832110286537, −3.54697916532804150617865558059, −2.18305808288351369047894609918, −0.857325965831403738730171627123,
0.857325965831403738730171627123, 2.18305808288351369047894609918, 3.54697916532804150617865558059, 4.36911497516141879832110286537, 5.27436486399365145320812776399, 5.97263589617596147119559165159, 6.42642510005325526840501326766, 8.040249755561486568046256570245, 8.593593995515546013560150047702, 9.496431189611638169176227558437