| L(s) = 1 | + (−0.984 − 0.173i)2-s + (1.34 + 1.09i)3-s + (0.939 + 0.342i)4-s + (2.60 − 2.18i)5-s + (−1.13 − 1.30i)6-s + (1.18 + 2.36i)7-s + (−0.866 − 0.5i)8-s + (0.616 + 2.93i)9-s + (−2.94 + 1.69i)10-s + (0.604 − 0.720i)11-s + (0.890 + 1.48i)12-s + (0.436 − 0.0769i)13-s + (−0.753 − 2.53i)14-s + (5.88 − 0.0950i)15-s + (0.766 + 0.642i)16-s + (−1.11 − 1.92i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 − 0.122i)2-s + (0.776 + 0.630i)3-s + (0.469 + 0.171i)4-s + (1.16 − 0.976i)5-s + (−0.463 − 0.534i)6-s + (0.446 + 0.894i)7-s + (−0.306 − 0.176i)8-s + (0.205 + 0.978i)9-s + (−0.929 + 0.536i)10-s + (0.182 − 0.217i)11-s + (0.256 + 0.428i)12-s + (0.121 − 0.0213i)13-s + (−0.201 − 0.677i)14-s + (1.51 − 0.0245i)15-s + (0.191 + 0.160i)16-s + (−0.269 − 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55066 + 0.217795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55066 + 0.217795i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 3 | \( 1 + (-1.34 - 1.09i)T \) |
| 7 | \( 1 + (-1.18 - 2.36i)T \) |
| good | 5 | \( 1 + (-2.60 + 2.18i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (-0.604 + 0.720i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.436 + 0.0769i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.11 + 1.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.85 + 3.95i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.171 - 0.470i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-8.00 - 1.41i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.68 - 7.37i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (0.250 + 0.433i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.16 + 6.61i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.79 - 3.18i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (2.17 - 0.790i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 7.36iT - 53T^{2} \) |
| 59 | \( 1 + (-0.00253 + 0.00213i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (4.17 + 11.4i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.67 - 15.1i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (10.3 - 5.98i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.87 - 2.23i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.93 + 10.9i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.34 + 13.2i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (4.96 - 8.60i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.02 - 1.21i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.10210875852608005233194034442, −10.26752208469111144518895170451, −9.293186419590988552750284828453, −8.792123721401644163989834563290, −8.318477658737538395286648138072, −6.70304531176080283546597875638, −5.44298530360082775478107301841, −4.56718389560642235209748961602, −2.73905438584388191501318456597, −1.74758132316670275864190571826,
1.58949184118842411674890613271, 2.55373641476773026480057045881, 4.08057081344383605169414417515, 6.15211286726391502093221031774, 6.61390756930966099228217258294, 7.65276784847183063850606325685, 8.446849838526515830370784098829, 9.544466460143723395362043737254, 10.31494961428085063554594638302, 10.92126823959628116523449362511
