| L(s) = 1 | + (5.65 + 9.79i)2-s + (−81.1 − 46.8i)3-s + (−63.9 + 110. i)4-s + (928. − 535. i)5-s − 1.06e3i·6-s + (2.21e3 − 931. i)7-s − 1.44e3·8-s + (1.11e3 + 1.92e3i)9-s + (1.05e4 + 6.06e3i)10-s + (312. − 541. i)11-s + (1.03e4 − 5.99e3i)12-s − 4.33e4i·13-s + (2.16e4 + 1.64e4i)14-s − 1.00e5·15-s + (−8.19e3 − 1.41e4i)16-s + (9.10e4 + 5.25e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−1.00 − 0.578i)3-s + (−0.249 + 0.433i)4-s + (1.48 − 0.857i)5-s − 0.818i·6-s + (0.921 − 0.388i)7-s − 0.353·8-s + (0.169 + 0.294i)9-s + (1.05 + 0.606i)10-s + (0.0213 − 0.0369i)11-s + (0.501 − 0.289i)12-s − 1.51i·13-s + (0.563 + 0.427i)14-s − 1.98·15-s + (−0.125 − 0.216i)16-s + (1.09 + 0.629i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.65463 - 0.441882i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65463 - 0.441882i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.65 - 9.79i)T \) |
| 7 | \( 1 + (-2.21e3 + 931. i)T \) |
| good | 3 | \( 1 + (81.1 + 46.8i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-928. + 535. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-312. + 541. i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 4.33e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-9.10e4 - 5.25e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.38e5 - 8.01e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.22e4 + 2.12e4i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 2.01e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-7.57e5 - 4.37e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-8.30e5 - 1.43e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 4.64e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.94e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (2.19e6 - 1.26e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (1.79e6 - 3.10e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-5.42e6 - 3.13e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (1.03e7 - 5.97e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (5.34e6 - 9.25e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.25e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-4.20e7 - 2.43e7i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.70e7 - 4.69e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 2.28e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-7.46e7 + 4.30e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.34e8iT - 7.83e15T^{2} \) |
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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
−17.30545163156010719148047306437, −16.84421808931269037751213111717, −14.69347566454627060504667973350, −13.26880383454072717850356254863, −12.33021810573450690334402437548, −10.34130895776376371127930086578, −8.204247710400935484621328944709, −6.11507787531486442945116852151, −5.19418556946566651612247671047, −1.18450388211463989832713757557,
2.08622226188505217827437433837, 4.91270221888839629060186733502, 6.23964846946892308889603243095, 9.527213724707014142327768658612, 10.75668464235490858723186448076, 11.72298931567224075154928013187, 13.75416801037365780639102680634, 14.74479503309559431488268775627, 16.77266418925755286829325182231, 17.81557464649403067181123006619
