| L(s) = 1 | + 8·2-s + 45.1i·3-s + 361. i·6-s + (−133 + 316. i)7-s − 512·8-s − 1.31e3·9-s + 874·11-s + 2.21e3i·13-s + (−1.06e3 + 2.52e3i)14-s − 4.09e3·16-s − 5.96e3i·17-s − 1.04e4·18-s − 3.11e3i·19-s + (−1.42e4 − 6.00e3i)21-s + 6.99e3·22-s − 4.73e3·23-s + ⋯ |
| L(s) = 1 | + 2-s + 1.67i·3-s + 1.67i·6-s + (−0.387 + 0.921i)7-s − 8-s − 1.79·9-s + 0.656·11-s + 1.00i·13-s + (−0.387 + 0.921i)14-s − 16-s − 1.21i·17-s − 1.79·18-s − 0.454i·19-s + (−1.54 − 0.648i)21-s + 0.656·22-s − 0.389·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.9669221547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9669221547\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + (133 - 316. i)T \) |
| good | 2 | \( 1 - 8T + 64T^{2} \) |
| 3 | \( 1 - 45.1iT - 729T^{2} \) |
| 11 | \( 1 - 874T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.21e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 5.96e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.11e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 4.73e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 1.11e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 2.74e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 3.00e3T + 2.56e9T^{2} \) |
| 41 | \( 1 + 5.75e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 3.14e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.24e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 7.64e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.13e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.75e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.95e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 1.84e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 6.09e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 5.34e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 7.14e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 6.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.14e5iT - 8.32e11T^{2} \) |
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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
−12.00959671199722806570489873783, −11.63778276199660885470575384476, −10.20522002894314212677244802597, −9.065600006469152950582992810981, −8.996211702847818874473736271095, −6.60895980633511322570869398468, −5.44483120114792004093139641267, −4.67886938816587593526536255492, −3.74442907001165738164534044211, −2.71748615879167177521774827043,
0.19737064908307279263389476548, 1.39987176172518001485823936364, 3.00802983564462121477512168469, 4.16161839509717405407014257039, 5.84592798493274590707823487794, 6.44327273022098385065769748672, 7.60364961117291651816559863889, 8.513078668637949097215147131721, 10.03227319433961134121385262846, 11.40900215205483730793360254123
