L(s) = 1 | + (−11.1 − 1.89i)2-s + (120. + 42.3i)4-s − 338. i·5-s − 438.·7-s + (−1.26e3 − 701. i)8-s + (−642. + 3.77e3i)10-s − 1.96e3i·11-s − 2.21e3i·13-s + (4.89e3 + 833. i)14-s + (1.27e4 + 1.02e4i)16-s + 1.21e4·17-s − 3.28e4i·19-s + (1.43e4 − 4.08e4i)20-s + (−3.73e3 + 2.19e4i)22-s − 1.96e4·23-s + ⋯ |
L(s) = 1 | + (−0.985 − 0.167i)2-s + (0.943 + 0.330i)4-s − 1.21i·5-s − 0.483·7-s + (−0.874 − 0.484i)8-s + (−0.203 + 1.19i)10-s − 0.445i·11-s − 0.279i·13-s + (0.476 + 0.0811i)14-s + (0.781 + 0.624i)16-s + 0.598·17-s − 1.09i·19-s + (0.400 − 1.14i)20-s + (−0.0747 + 0.439i)22-s − 0.335·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.0658226 + 0.254754i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0658226 + 0.254754i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (11.1 + 1.89i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 338. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 438.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.96e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 2.21e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.21e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.28e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 1.96e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.60e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 2.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.96e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 5.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.83e4iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 8.20e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.53e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.82e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 4.84e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 7.98e4iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.70e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.53e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 1.99e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.89e4T + 8.07e13T^{2} \) |
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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
−12.45182665742906201604716346386, −11.36116936633452887914458142122, −10.05907481417222570861195057053, −9.035154969557980494528495601913, −8.220493579278695165927205597653, −6.78198619768351734910138602511, −5.23970764719000336493156881554, −3.25508714501174123310737236140, −1.38322821644698460879146032104, −0.12683193950122045422149341679,
1.95501909187412451312811374304, 3.40715880262284872740108525652, 5.90527186896910653998969792303, 6.94556965670812110472321841823, 7.927431290486798793049971239412, 9.508394072196889773100326914770, 10.28429610829567998813756365991, 11.29612695150281477225147519187, 12.46384923180672771863004911874, 14.21587100772732369902184546830