| L(s) = 1 | + (40.7 − 70.6i)2-s + (482. − 835. i)3-s + (767. + 1.32e3i)4-s + 4.15e4·5-s + (−3.93e4 − 6.81e4i)6-s + (1.58e5 + 2.74e5i)7-s + 7.93e5·8-s + (3.32e5 + 5.75e5i)9-s + (1.69e6 − 2.93e6i)10-s + (9.70e5 − 1.68e6i)11-s + 1.48e6·12-s + (−1.60e7 + 6.66e6i)13-s + 2.58e7·14-s + (2.00e7 − 3.46e7i)15-s + (2.60e7 − 4.51e7i)16-s + (−3.77e7 − 6.54e7i)17-s + ⋯ |
| L(s) = 1 | + (0.450 − 0.780i)2-s + (0.381 − 0.661i)3-s + (0.0936 + 0.162i)4-s + 1.18·5-s + (−0.344 − 0.596i)6-s + (0.508 + 0.881i)7-s + 1.07·8-s + (0.208 + 0.360i)9-s + (0.535 − 0.927i)10-s + (0.165 − 0.285i)11-s + 0.143·12-s + (−0.923 + 0.383i)13-s + 0.917·14-s + (0.453 − 0.785i)15-s + (0.388 − 0.673i)16-s + (−0.379 − 0.657i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.18260 - 1.58106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.18260 - 1.58106i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.60e7 - 6.66e6i)T \) |
| good | 2 | \( 1 + (-40.7 + 70.6i)T + (-4.09e3 - 7.09e3i)T^{2} \) |
| 3 | \( 1 + (-482. + 835. i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 - 4.15e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + (-1.58e5 - 2.74e5i)T + (-4.84e10 + 8.39e10i)T^{2} \) |
| 11 | \( 1 + (-9.70e5 + 1.68e6i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 17 | \( 1 + (3.77e7 + 6.54e7i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (8.28e7 + 1.43e8i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-3.83e8 + 6.64e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 + (1.97e9 - 3.41e9i)T + (-5.13e18 - 8.88e18i)T^{2} \) |
| 31 | \( 1 + 9.28e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + (5.00e9 - 8.66e9i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 + (-2.24e10 + 3.88e10i)T + (-4.62e20 - 8.01e20i)T^{2} \) |
| 43 | \( 1 + (-7.71e9 - 1.33e10i)T + (-8.59e20 + 1.48e21i)T^{2} \) |
| 47 | \( 1 + 1.68e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.02e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + (-2.12e10 - 3.67e10i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (2.84e11 + 4.93e11i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (2.99e11 - 5.18e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + (4.95e11 + 8.58e11i)T + (-5.82e23 + 1.00e24i)T^{2} \) |
| 73 | \( 1 + 1.38e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 5.45e10T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.21e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (3.08e12 - 5.35e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + (7.80e11 + 1.35e12i)T + (-3.36e25 + 5.82e25i)T^{2} \) |
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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
−16.60755111496863774659666956630, −14.46707937164464077444639159803, −13.35653844704879556815722075184, −12.37544977672010461207895586509, −10.86252401135677212638124992106, −9.002716253736338840864091798883, −7.14405481598639910075898837873, −5.03415420031634948869825189376, −2.51872860266119969161299806103, −1.82246519509805106462422654793,
1.67263758614322874647253437743, 4.23920536042220205368957141362, 5.74843510858468675410991839189, 7.35027249358671760371351077850, 9.582470020518718624996330955659, 10.59781920178811222849631083552, 13.15460729844261421678741264284, 14.42484735199808254746004533892, 15.08206787670440473467648254846, 16.71970329219784699549121246507
