| L(s) = 1 | + (−25.0 + 43.3i)2-s + (227. − 131. i)3-s + (792. + 1.37e3i)4-s + (6.61e3 + 3.82e3i)5-s + 1.31e4i·6-s + (−1.06e5 + 5.05e4i)7-s − 2.84e5·8-s + (−2.31e5 + 4.00e5i)9-s + (−3.31e5 + 1.91e5i)10-s + (4.18e5 + 7.24e5i)11-s + (3.60e5 + 2.08e5i)12-s + 8.22e5i·13-s + (4.65e5 − 5.87e6i)14-s + 2.00e6·15-s + (3.88e6 − 6.73e6i)16-s + (1.13e7 − 6.57e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.391 + 0.678i)2-s + (0.312 − 0.180i)3-s + (0.193 + 0.335i)4-s + (0.423 + 0.244i)5-s + 0.282i·6-s + (−0.902 + 0.430i)7-s − 1.08·8-s + (−0.434 + 0.753i)9-s + (−0.331 + 0.191i)10-s + (0.236 + 0.408i)11-s + (0.120 + 0.0697i)12-s + 0.170i·13-s + (0.0618 − 0.780i)14-s + 0.176·15-s + (0.231 − 0.401i)16-s + (0.471 − 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.445414 + 1.16661i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.445414 + 1.16661i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.06e5 - 5.05e4i)T \) |
| good | 2 | \( 1 + (25.0 - 43.3i)T + (-2.04e3 - 3.54e3i)T^{2} \) |
| 3 | \( 1 + (-227. + 131. i)T + (2.65e5 - 4.60e5i)T^{2} \) |
| 5 | \( 1 + (-6.61e3 - 3.82e3i)T + (1.22e8 + 2.11e8i)T^{2} \) |
| 11 | \( 1 + (-4.18e5 - 7.24e5i)T + (-1.56e12 + 2.71e12i)T^{2} \) |
| 13 | \( 1 - 8.22e5iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (-1.13e7 + 6.57e6i)T + (2.91e14 - 5.04e14i)T^{2} \) |
| 19 | \( 1 + (-3.50e7 - 2.02e7i)T + (1.10e15 + 1.91e15i)T^{2} \) |
| 23 | \( 1 + (6.36e7 - 1.10e8i)T + (-1.09e16 - 1.89e16i)T^{2} \) |
| 29 | \( 1 - 5.91e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (-1.40e9 + 8.13e8i)T + (3.93e17 - 6.82e17i)T^{2} \) |
| 37 | \( 1 + (-1.21e8 + 2.09e8i)T + (-3.29e18 - 5.70e18i)T^{2} \) |
| 41 | \( 1 - 6.47e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 - 7.74e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (4.02e9 + 2.32e9i)T + (5.80e19 + 1.00e20i)T^{2} \) |
| 53 | \( 1 + (-6.84e9 - 1.18e10i)T + (-2.45e20 + 4.25e20i)T^{2} \) |
| 59 | \( 1 + (5.55e10 - 3.20e10i)T + (8.89e20 - 1.54e21i)T^{2} \) |
| 61 | \( 1 + (7.42e10 + 4.28e10i)T + (1.32e21 + 2.29e21i)T^{2} \) |
| 67 | \( 1 + (2.89e10 + 5.01e10i)T + (-4.09e21 + 7.08e21i)T^{2} \) |
| 71 | \( 1 - 1.65e11T + 1.64e22T^{2} \) |
| 73 | \( 1 + (2.24e9 - 1.29e9i)T + (1.14e22 - 1.98e22i)T^{2} \) |
| 79 | \( 1 + (8.57e10 - 1.48e11i)T + (-2.95e22 - 5.11e22i)T^{2} \) |
| 83 | \( 1 - 5.37e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (-5.34e11 - 3.08e11i)T + (1.23e23 + 2.13e23i)T^{2} \) |
| 97 | \( 1 + 1.46e12iT - 6.93e23T^{2} \) |
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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
−19.75135584992037640593510538531, −18.28367459781231367073141336319, −16.84791147678506619682278527187, −15.64075922681106573842711412277, −13.87304469610041092563785483158, −12.06770852819054233151239134737, −9.609401952121066300489781929304, −7.87127422560107197993270042541, −6.17102715542255571219787793033, −2.75235789920686313343451377872,
0.77369066424433264000811170709, 3.08882363271621237090797004181, 6.19703523322209825484288475742, 9.093986667738357163173657387479, 10.27188036942112908804629019661, 12.12479778554122393388791823750, 14.04302885931622438388100802436, 15.74032818181149238895957093254, 17.52750803357683916611145215637, 19.17694182892075950054413937230
