L(s) = 1 | + 2.56i·3-s − 3.56·9-s + 5.12·11-s − 4.56i·13-s − 3.12i·17-s − 5.12·19-s + i·23-s − 1.43i·27-s + 0.561·29-s − 6.56·31-s + 13.1i·33-s − 8.24i·37-s + 11.6·39-s + 10.8·41-s + 8i·43-s + ⋯ |
L(s) = 1 | + 1.47i·3-s − 1.18·9-s + 1.54·11-s − 1.26i·13-s − 0.757i·17-s − 1.17·19-s + 0.208i·23-s − 0.276i·27-s + 0.104·29-s − 1.17·31-s + 2.28i·33-s − 1.35i·37-s + 1.87·39-s + 1.68·41-s + 1.21i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.025746643\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.025746643\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 2.56iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 + 4.56iT - 13T^{2} \) |
| 17 | \( 1 + 3.12iT - 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 29 | \( 1 - 0.561T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 + 8.24iT - 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 5.12iT - 67T^{2} \) |
| 71 | \( 1 - 9.43T + 71T^{2} \) |
| 73 | \( 1 - 2.31iT - 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 - 2.24iT - 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 13.3iT - 97T^{2} \) |
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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
−8.696383306322974671358961021601, −7.82992079831242737983847822265, −6.98073231382581887409942477445, −6.03914232563981520678430201985, −5.46396278588778067804249992353, −4.55329600351763514626853545604, −3.98123345621833431160936344067, −3.35194577839899351848530003943, −2.31593614068389934968889262050, −0.800811225401481398194365439291,
0.814681270084052902003881604503, 1.81316568199707686160866374596, 2.24743638525632042165945964563, 3.77619244735410390382509217800, 4.21486758469910550504300723464, 5.51122781450979889972416021916, 6.39493552824126593723961247281, 6.70418368249559564078088879173, 7.23161536392081926461291779014, 8.223194188839842677286287051048