L(s) = 1 | + 2.44·2-s − 5i·3-s − 10·4-s − 12.2i·6-s + (34.2 + 35i)7-s − 63.6·8-s + 56·9-s + 89·11-s + 50i·12-s + 5i·13-s + (84 + 85.7i)14-s + 4.00·16-s − 485i·17-s + 137.·18-s − 220. i·19-s + ⋯ |
L(s) = 1 | + 0.612·2-s − 0.555i·3-s − 0.625·4-s − 0.340i·6-s + (0.699 + 0.714i)7-s − 0.995·8-s + 0.691·9-s + 0.735·11-s + 0.347i·12-s + 0.0295i·13-s + (0.428 + 0.437i)14-s + 0.0156·16-s − 1.67i·17-s + 0.423·18-s − 0.610i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.389306038\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.389306038\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (-34.2 - 35i)T \) |
good | 2 | \( 1 - 2.44T + 16T^{2} \) |
| 3 | \( 1 + 5iT - 81T^{2} \) |
| 11 | \( 1 - 89T + 1.46e4T^{2} \) |
| 13 | \( 1 - 5iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 485iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 220. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 700.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 191T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.05e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.63e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.91e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 377.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.19e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.58e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.62e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.93e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.04e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.45e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 8.65e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 5.56e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.99e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 808. iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.23e3iT - 8.85e7T^{2} \) |
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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
−11.97311118856286828449977060523, −11.39876953974403929440713606644, −9.617070871189276432702654823701, −8.963025758518572956254440134520, −7.68571801524337607419092419298, −6.53735647500537441695169472276, −5.21928027558551717417832487207, −4.35578366080234831855500341990, −2.68921841071229965522198422668, −0.931680428209438642526386907378,
1.28991936313571238717153188155, 3.67762951699722141021849825910, 4.29050093660580788770182697835, 5.38186072607895411461978962946, 6.77954704677749980208412713131, 8.169731039020927395243762209300, 9.186139502440352297085993828250, 10.23215689517568391671925206483, 11.08329246106042865290814557630, 12.44128161900203611555197971466