L(s) = 1 | + (1.96 − 0.525i)2-s + (−0.866 − 0.5i)3-s + (1.84 − 1.06i)4-s + (1.56 − 1.56i)5-s + (−1.96 − 0.525i)6-s + (2.02 + 1.70i)7-s + (0.187 − 0.187i)8-s + (0.499 + 0.866i)9-s + (2.25 − 3.90i)10-s + (−1.16 − 4.33i)11-s − 2.13·12-s + (−3.51 − 0.786i)13-s + (4.87 + 2.27i)14-s + (−2.14 + 0.573i)15-s + (−1.86 + 3.22i)16-s + (1.31 + 2.27i)17-s + ⋯ |
L(s) = 1 | + (1.38 − 0.371i)2-s + (−0.499 − 0.288i)3-s + (0.922 − 0.532i)4-s + (0.700 − 0.700i)5-s + (−0.801 − 0.214i)6-s + (0.765 + 0.643i)7-s + (0.0661 − 0.0661i)8-s + (0.166 + 0.288i)9-s + (0.712 − 1.23i)10-s + (−0.350 − 1.30i)11-s − 0.614·12-s + (−0.975 − 0.218i)13-s + (1.30 + 0.609i)14-s + (−0.552 + 0.148i)15-s + (−0.465 + 0.805i)16-s + (0.318 + 0.552i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.15513 - 1.03236i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.15513 - 1.03236i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.02 - 1.70i)T \) |
| 13 | \( 1 + (3.51 + 0.786i)T \) |
good | 2 | \( 1 + (-1.96 + 0.525i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-1.56 + 1.56i)T - 5iT^{2} \) |
| 11 | \( 1 + (1.16 + 4.33i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.31 - 2.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.01 - 1.61i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (4.58 + 2.64i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.06 - 3.57i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.44 - 2.44i)T - 31iT^{2} \) |
| 37 | \( 1 + (-0.0290 - 0.108i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.03 - 7.58i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (5.47 - 3.16i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.82 + 5.82i)T + 47iT^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 + (-1.27 + 4.77i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (3.33 - 1.92i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.1 + 2.72i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-3.56 + 13.3i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (11.8 + 11.8i)T + 73iT^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + (-5.15 + 5.15i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.93 + 2.12i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (1.93 + 0.518i)T + (84.0 + 48.5i)T^{2} \) |
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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
−11.95273069058714734122989316154, −11.32531531000104866510816824669, −10.17449205148119830051890265173, −8.853365901165179121433731933684, −7.84904878011570252527678173470, −6.12056539230552895608089446329, −5.42436105186123816574114497193, −4.90370722809489386409041147720, −3.22197408056256453416228030992, −1.77086811832289655966851983600,
2.38793767213597525070182325936, 3.99119929511605375667997649136, 4.97777468346294954628942065029, 5.65660546490820686962068901678, 7.03376656248193247302778194650, 7.44278499735365580968923460751, 9.694429025455858873509440371705, 10.07010113424732286278689398714, 11.46537055638381736766079701103, 12.04515493993550936713538320818