| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 2.37i·7-s + i·8-s − 9-s + 4.37·11-s − i·12-s + 2i·13-s + 2.37·14-s + 16-s + i·17-s + i·18-s + 2.37·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.896i·7-s + 0.353i·8-s − 0.333·9-s + 1.31·11-s − 0.288i·12-s + 0.554i·13-s + 0.634·14-s + 0.250·16-s + 0.242i·17-s + 0.235i·18-s + 0.544·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593115242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593115242\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 - 2.37iT - 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 1.37iT - 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 - 9.11T + 31T^{2} \) |
| 37 | \( 1 - iT - 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 - 3.62iT - 43T^{2} \) |
| 47 | \( 1 - 1.62iT - 47T^{2} \) |
| 53 | \( 1 - 5.74iT - 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 8.11T + 61T^{2} \) |
| 67 | \( 1 + 0.372iT - 67T^{2} \) |
| 71 | \( 1 + 1.37T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 1.37iT - 83T^{2} \) |
| 89 | \( 1 - 2.74T + 89T^{2} \) |
| 97 | \( 1 - 12.7iT - 97T^{2} \) |
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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
−9.307307407276683305065061670229, −8.604682897531078808502983777527, −7.70188036324846978730625200231, −6.51312430812791634107078946625, −5.88366384354115777490363533050, −4.89936652701205194463596356839, −4.15586574744672136747092545990, −3.35432309237983197299524669203, −2.37645657570172212763872286168, −1.29503980217968573601059960113,
0.58728185645740419556385763219, 1.63427384201488568092796732470, 3.20336107258129022956583148183, 3.98471026516833253346243122353, 4.91236418282782427833711968368, 5.89139532141344706614779335656, 6.53526738545254798820736633445, 7.31279200772717277584360521177, 7.71858806857016771239466920107, 8.657720895060923816766697434000
