L(s) = 1 | + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (0.781 − 0.623i)8-s + (−0.733 + 0.680i)11-s + (−0.680 − 0.733i)13-s + (0.623 − 0.781i)16-s + (−0.563 − 0.826i)17-s + (−0.5 − 0.866i)19-s + (−0.563 + 0.826i)22-s + (−0.997 − 0.0747i)23-s + (−0.826 − 0.563i)26-s + (−0.826 + 0.563i)29-s − 31-s + (0.433 − 0.900i)32-s + (−0.733 − 0.680i)34-s + ⋯ |
L(s) = 1 | + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (0.781 − 0.623i)8-s + (−0.733 + 0.680i)11-s + (−0.680 − 0.733i)13-s + (0.623 − 0.781i)16-s + (−0.563 − 0.826i)17-s + (−0.5 − 0.866i)19-s + (−0.563 + 0.826i)22-s + (−0.997 − 0.0747i)23-s + (−0.826 − 0.563i)26-s + (−0.826 + 0.563i)29-s − 31-s + (0.433 − 0.900i)32-s + (−0.733 − 0.680i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07817501614 - 1.032912773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07817501614 - 1.032912773i\) |
\(L(1)\) |
\(\approx\) |
\(1.324681731 - 0.4281799888i\) |
\(L(1)\) |
\(\approx\) |
\(1.324681731 - 0.4281799888i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.974 - 0.222i)T \) |
| 11 | \( 1 + (-0.733 + 0.680i)T \) |
| 13 | \( 1 + (-0.680 - 0.733i)T \) |
| 17 | \( 1 + (-0.563 - 0.826i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.997 - 0.0747i)T \) |
| 29 | \( 1 + (-0.826 + 0.563i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.997 + 0.0747i)T \) |
| 41 | \( 1 + (-0.365 + 0.930i)T \) |
| 43 | \( 1 + (0.930 - 0.365i)T \) |
| 47 | \( 1 + (-0.974 + 0.222i)T \) |
| 53 | \( 1 + (-0.997 - 0.0747i)T \) |
| 59 | \( 1 + (0.623 - 0.781i)T \) |
| 61 | \( 1 + (0.900 + 0.433i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (-0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.680 - 0.733i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.680 - 0.733i)T \) |
| 89 | \( 1 + (0.955 - 0.294i)T \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.25615514214933477134635012612, −19.37009761400712315436912106016, −18.85707324328321236916165462802, −17.742674407008851466438482917453, −16.98482629324236336222054581806, −16.33615340727723118248028792348, −15.711688019886257872094462523770, −14.88509752883405657039378284867, −14.28251707277087844115073178110, −13.595485875047069094030176246325, −12.8152833515589864276857508479, −12.25975707117066216769127978594, −11.38183925426968337726220544329, −10.73775907943837419543055036750, −9.95910466153233242003419550407, −8.78820188131449214318300223879, −8.01467399272526710898474869786, −7.33085008060832126406974762739, −6.38029231194815948810827603477, −5.77883648560632389877097913218, −4.995255921868521932493678420, −4.02905479045773008728513424122, −3.51751911984990804223741085706, −2.275539584372060897425845856379, −1.80944555381008250745283789886,
0.20023385839005410321025461703, 1.76955893214284383603172155166, 2.46211928481920571727456778075, 3.22778956570135076549282834101, 4.26678083490930070037351403328, 5.007271173310597834479304632910, 5.51908937862594803887828001771, 6.63666604496433126259234289710, 7.26212222692852599278214422983, 7.97883687788492916542416766201, 9.20611973968375493522073496739, 10.02026772666274368194825896345, 10.68689336137884709470408943691, 11.43015832071888053083276696520, 12.22522119453311213909377478073, 12.986080625059266837427581459072, 13.31415196164276410160672659560, 14.464790270584386247336224727783, 14.83731421020768942536406120038, 15.79643287824275476657661102318, 16.08384179877344770663153173461, 17.298471652455413474519230711590, 17.90369061083919881111047397291, 18.79314197593221934430538548987, 19.66931093892863653399553156881