L(s) = 1 | − 16·2-s + 256·4-s + 684·5-s + 9.14e3·7-s − 4.09e3·8-s − 1.09e4·10-s − 5.79e3·11-s − 1.79e5·13-s − 1.46e5·14-s + 6.55e4·16-s + 5.94e5·17-s + 1.30e5·19-s + 1.75e5·20-s + 9.26e4·22-s + 1.74e6·23-s − 1.48e6·25-s + 2.87e6·26-s + 2.34e6·28-s − 4.31e6·29-s + 1.60e5·31-s − 1.04e6·32-s − 9.50e6·34-s + 6.25e6·35-s − 2.19e7·37-s − 2.08e6·38-s − 2.80e6·40-s − 2.94e5·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.489·5-s + 1.44·7-s − 0.353·8-s − 0.346·10-s − 0.119·11-s − 1.74·13-s − 1.01·14-s + 1/4·16-s + 1.72·17-s + 0.229·19-s + 0.244·20-s + 0.0843·22-s + 1.30·23-s − 0.760·25-s + 1.23·26-s + 0.720·28-s − 1.13·29-s + 0.0311·31-s − 0.176·32-s − 1.21·34-s + 0.704·35-s − 1.92·37-s − 0.162·38-s − 0.173·40-s − 0.0162·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{4} T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - p^{4} T \) |
good | 5 | \( 1 - 684 T + p^{9} T^{2} \) |
| 7 | \( 1 - 1307 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 5790 T + p^{9} T^{2} \) |
| 13 | \( 1 + 13837 p T + p^{9} T^{2} \) |
| 17 | \( 1 - 594093 T + p^{9} T^{2} \) |
| 23 | \( 1 - 1744767 T + p^{9} T^{2} \) |
| 29 | \( 1 + 4314387 T + p^{9} T^{2} \) |
| 31 | \( 1 - 160232 T + p^{9} T^{2} \) |
| 37 | \( 1 + 21943090 T + p^{9} T^{2} \) |
| 41 | \( 1 + 294816 T + p^{9} T^{2} \) |
| 43 | \( 1 + 39393148 T + p^{9} T^{2} \) |
| 47 | \( 1 + 46596360 T + p^{9} T^{2} \) |
| 53 | \( 1 + 22121703 T + p^{9} T^{2} \) |
| 59 | \( 1 + 33070233 T + p^{9} T^{2} \) |
| 61 | \( 1 - 188535938 T + p^{9} T^{2} \) |
| 67 | \( 1 + 20769067 T + p^{9} T^{2} \) |
| 71 | \( 1 - 232299978 T + p^{9} T^{2} \) |
| 73 | \( 1 + 3022183 T + p^{9} T^{2} \) |
| 79 | \( 1 + 446379406 T + p^{9} T^{2} \) |
| 83 | \( 1 + 794022846 T + p^{9} T^{2} \) |
| 89 | \( 1 + 90999336 T + p^{9} T^{2} \) |
| 97 | \( 1 + 123974170 T + p^{9} T^{2} \) |
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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
−9.698496549674461351891327520698, −8.520465144713392807442996220049, −7.69104812560962385631423318346, −7.03934618671628664335909150072, −5.42547721784552739347623875481, −4.99786974778475595398451503809, −3.28072028515980892315762027438, −2.03229054879918073889335982040, −1.34448136559822445809810865722, 0,
1.34448136559822445809810865722, 2.03229054879918073889335982040, 3.28072028515980892315762027438, 4.99786974778475595398451503809, 5.42547721784552739347623875481, 7.03934618671628664335909150072, 7.69104812560962385631423318346, 8.520465144713392807442996220049, 9.698496549674461351891327520698