L(s) = 1 | + (−56.2 − 70.9i)2-s + (1.62e3 + 1.62e3i)3-s + (−1.86e3 + 7.97e3i)4-s + (8.03e3 − 8.03e3i)5-s + (2.37e4 − 2.06e5i)6-s − 5.16e5i·7-s + (6.70e5 − 3.16e5i)8-s + 3.66e6i·9-s + (−1.02e6 − 1.17e5i)10-s + (4.57e6 − 4.57e6i)11-s + (−1.59e7 + 9.91e6i)12-s + (1.23e7 + 1.23e7i)13-s + (−3.66e7 + 2.90e7i)14-s + 2.60e7·15-s + (−6.01e7 − 2.97e7i)16-s + 4.11e7·17-s + ⋯ |
L(s) = 1 | + (−0.621 − 0.783i)2-s + (1.28 + 1.28i)3-s + (−0.227 + 0.973i)4-s + (0.230 − 0.230i)5-s + (0.208 − 1.80i)6-s − 1.65i·7-s + (0.904 − 0.426i)8-s + 2.29i·9-s + (−0.323 − 0.0372i)10-s + (0.778 − 0.778i)11-s + (−1.54 + 0.958i)12-s + (0.707 + 0.707i)13-s + (−1.29 + 1.03i)14-s + 0.590·15-s + (−0.896 − 0.443i)16-s + 0.413·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0667i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(2.32452 - 0.0776833i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.32452 - 0.0776833i\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (56.2 + 70.9i)T \) |
good | 3 | \( 1 + (-1.62e3 - 1.62e3i)T + 1.59e6iT^{2} \) |
| 5 | \( 1 + (-8.03e3 + 8.03e3i)T - 1.22e9iT^{2} \) |
| 7 | \( 1 + 5.16e5iT - 9.68e10T^{2} \) |
| 11 | \( 1 + (-4.57e6 + 4.57e6i)T - 3.45e13iT^{2} \) |
| 13 | \( 1 + (-1.23e7 - 1.23e7i)T + 3.02e14iT^{2} \) |
| 17 | \( 1 - 4.11e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + (-2.09e8 - 2.09e8i)T + 4.20e16iT^{2} \) |
| 23 | \( 1 + 3.35e8iT - 5.04e17T^{2} \) |
| 29 | \( 1 + (1.25e9 + 1.25e9i)T + 1.02e19iT^{2} \) |
| 31 | \( 1 - 9.15e8T + 2.44e19T^{2} \) |
| 37 | \( 1 + (-9.04e9 + 9.04e9i)T - 2.43e20iT^{2} \) |
| 41 | \( 1 - 2.15e10iT - 9.25e20T^{2} \) |
| 43 | \( 1 + (-3.26e9 + 3.26e9i)T - 1.71e21iT^{2} \) |
| 47 | \( 1 - 7.63e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + (-5.41e10 + 5.41e10i)T - 2.60e22iT^{2} \) |
| 59 | \( 1 + (-4.71e10 + 4.71e10i)T - 1.04e23iT^{2} \) |
| 61 | \( 1 + (3.10e11 + 3.10e11i)T + 1.61e23iT^{2} \) |
| 67 | \( 1 + (9.94e11 + 9.94e11i)T + 5.48e23iT^{2} \) |
| 71 | \( 1 - 1.34e12iT - 1.16e24T^{2} \) |
| 73 | \( 1 + 1.99e11iT - 1.67e24T^{2} \) |
| 79 | \( 1 + 1.43e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + (2.07e10 + 2.07e10i)T + 8.87e24iT^{2} \) |
| 89 | \( 1 + 3.61e12iT - 2.19e25T^{2} \) |
| 97 | \( 1 + 4.73e12T + 6.73e25T^{2} \) |
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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
−16.32097651751526809889820315525, −14.20176334521093122599186342616, −13.55381505378130778815573781217, −11.13321873493229775063456338063, −10.02570289556749282337640975928, −9.068654087113676332664416891793, −7.75102273278918028156712275511, −4.14308120799364302752445491742, −3.42129634382353350195806128556, −1.29999159577686646521544258841,
1.29383318829989463615642335733, 2.62743350357943623222375423233, 5.93407597314211805528768687699, 7.27131140858816118397082304905, 8.575334828218287391034745459981, 9.413109557491814498974438368818, 12.08127780428973325419873742920, 13.53895773969813305711906142830, 14.72427997359297086787983572932, 15.51859122396886430846017323746