| L(s) = 1 | + (−41.6 + 72.0i)2-s + (−951. − 1.64e3i)3-s + (633. + 1.09e3i)4-s + (8.42e3 − 1.45e4i)5-s + 1.58e5·6-s + (9.29e4 + 2.97e5i)7-s − 7.87e5·8-s + (−1.01e6 + 1.75e6i)9-s + (7.00e5 + 1.21e6i)10-s + (4.91e6 + 8.52e6i)11-s + (1.20e6 − 2.08e6i)12-s + 1.62e7·13-s + (−2.52e7 − 5.65e6i)14-s − 3.20e7·15-s + (2.75e7 − 4.77e7i)16-s + (3.27e7 + 5.67e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.459 + 0.796i)2-s + (−0.753 − 1.30i)3-s + (0.0773 + 0.133i)4-s + (0.241 − 0.417i)5-s + 1.38·6-s + (0.298 + 0.954i)7-s − 1.06·8-s + (−0.635 + 1.10i)9-s + (0.221 + 0.383i)10-s + (0.837 + 1.45i)11-s + (0.116 − 0.201i)12-s + 0.936·13-s + (−0.897 − 0.200i)14-s − 0.726·15-s + (0.410 − 0.711i)16-s + (0.329 + 0.570i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.773791 + 0.661020i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.773791 + 0.661020i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-9.29e4 - 2.97e5i)T \) |
| good | 2 | \( 1 + (41.6 - 72.0i)T + (-4.09e3 - 7.09e3i)T^{2} \) |
| 3 | \( 1 + (951. + 1.64e3i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-8.42e3 + 1.45e4i)T + (-6.10e8 - 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-4.91e6 - 8.52e6i)T + (-1.72e13 + 2.98e13i)T^{2} \) |
| 13 | \( 1 - 1.62e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-3.27e7 - 5.67e7i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (1.62e7 - 2.80e7i)T + (-2.10e16 - 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-1.71e8 + 2.97e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 + 4.72e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (-2.14e9 - 3.72e9i)T + (-1.22e19 + 2.11e19i)T^{2} \) |
| 37 | \( 1 + (9.22e9 - 1.59e10i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 + 6.55e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 6.12e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-5.58e10 + 9.67e10i)T + (-2.73e21 - 4.72e21i)T^{2} \) |
| 53 | \( 1 + (2.45e10 + 4.25e10i)T + (-1.30e22 + 2.25e22i)T^{2} \) |
| 59 | \( 1 + (-1.43e11 - 2.48e11i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (3.11e11 - 5.38e11i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-1.36e11 - 2.36e11i)T + (-2.74e23 + 4.74e23i)T^{2} \) |
| 71 | \( 1 + 5.54e10T + 1.16e24T^{2} \) |
| 73 | \( 1 + (4.97e11 + 8.61e11i)T + (-8.35e23 + 1.44e24i)T^{2} \) |
| 79 | \( 1 + (-5.68e11 + 9.84e11i)T + (-2.33e24 - 4.04e24i)T^{2} \) |
| 83 | \( 1 + 4.20e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (-2.05e12 + 3.55e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 - 1.00e13T + 6.73e25T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.73696773027828362190378938401, −17.74503383269060033169414172979, −16.88094021813343025941562809597, −15.08661211017158423634591020823, −12.74157766711403730034271797733, −11.84979283571041174502197544740, −8.796932807954755427707704349408, −7.15492794893823988646869810427, −5.85458471942868784902155171724, −1.60316346897584381927516710492,
0.74458530033148120765973851830, 3.66351084635938659521501196105, 5.95508147394050141290465004431, 9.337476247167194228021085442937, 10.79453555842300747422019349911, 11.25792602191015711839675325632, 14.20892277913696648056203337657, 15.95162529182532722165782625428, 17.22222247312646548145715104988, 18.86049646888844501825809967870
