L(s) = 1 | − 0.0898i·2-s + i·3-s + 1.99·4-s + 0.0898·6-s + 4.36i·7-s − 0.358i·8-s − 9-s + 4.39·11-s + 1.99i·12-s + 1.98i·13-s + 0.391·14-s + 3.95·16-s + 0.997i·17-s + 0.0898i·18-s − 1.35·19-s + ⋯ |
L(s) = 1 | − 0.0635i·2-s + 0.577i·3-s + 0.995·4-s + 0.0366·6-s + 1.64i·7-s − 0.126i·8-s − 0.333·9-s + 1.32·11-s + 0.575i·12-s + 0.549i·13-s + 0.104·14-s + 0.987·16-s + 0.241i·17-s + 0.0211i·18-s − 0.309·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.360464941\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.360464941\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 0.0898iT - 2T^{2} \) |
| 7 | \( 1 - 4.36iT - 7T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 13 | \( 1 - 1.98iT - 13T^{2} \) |
| 17 | \( 1 - 0.997iT - 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 23 | \( 1 + 2.35iT - 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 + 3.67T + 31T^{2} \) |
| 37 | \( 1 + 1.43iT - 37T^{2} \) |
| 41 | \( 1 + 5.98T + 41T^{2} \) |
| 43 | \( 1 + 2.68iT - 43T^{2} \) |
| 47 | \( 1 - 10.9iT - 47T^{2} \) |
| 53 | \( 1 + 11.0iT - 53T^{2} \) |
| 59 | \( 1 + 6.68T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 - 12.9iT - 67T^{2} \) |
| 71 | \( 1 - 7.32T + 71T^{2} \) |
| 73 | \( 1 - 0.424iT - 73T^{2} \) |
| 79 | \( 1 - 6.35T + 79T^{2} \) |
| 83 | \( 1 + 0.737iT - 83T^{2} \) |
| 89 | \( 1 - 9.78T + 89T^{2} \) |
| 97 | \( 1 - 0.0337iT - 97T^{2} \) |
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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
−9.314923885700460834007407465494, −8.793691442116926391393080759781, −8.039734498971952547897688852890, −6.72950592920919142799440558586, −6.32853420077471931877083677456, −5.52406029160529845990673252021, −4.50490089076100458497674586668, −3.42938706387538025976643467449, −2.52552978681891474943384107472, −1.62552315832980889346852496148,
0.887606847308669710052280881025, 1.72733939672791954151113898787, 3.11337912834457163866977951110, 3.86171754776406903080667511288, 4.98107650752665993856903368305, 6.29328976244314287339313678528, 6.63111402360947756535292676348, 7.42016977323970355096749334964, 7.909183304195615313312195937649, 8.955050549820015677095823994961