| L(s) = 1 | + (−0.399 − 1.35i)2-s + (1.62 − 2.11i)3-s + (−1.68 + 1.08i)4-s + (1.05 − 0.807i)5-s + (−3.51 − 1.35i)6-s + (2.64 − 0.0808i)7-s + (2.14 + 1.84i)8-s + (−1.05 − 3.94i)9-s + (−1.51 − 1.10i)10-s + (−1.69 + 0.223i)11-s + (−0.432 + 5.30i)12-s + (−0.550 + 0.228i)13-s + (−1.16 − 3.55i)14-s − 3.53i·15-s + (1.64 − 3.64i)16-s + (−4.66 + 2.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.282 − 0.959i)2-s + (0.935 − 1.21i)3-s + (−0.840 + 0.542i)4-s + (0.470 − 0.361i)5-s + (−1.43 − 0.552i)6-s + (0.999 − 0.0305i)7-s + (0.757 + 0.652i)8-s + (−0.352 − 1.31i)9-s + (−0.479 − 0.349i)10-s + (−0.511 + 0.0673i)11-s + (−0.124 + 1.53i)12-s + (−0.152 + 0.0632i)13-s + (−0.311 − 0.950i)14-s − 0.911i·15-s + (0.411 − 0.911i)16-s + (−1.13 + 0.652i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.652957 - 1.28884i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.652957 - 1.28884i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.399 + 1.35i)T \) |
| 7 | \( 1 + (-2.64 + 0.0808i)T \) |
| good | 3 | \( 1 + (-1.62 + 2.11i)T + (-0.776 - 2.89i)T^{2} \) |
| 5 | \( 1 + (-1.05 + 0.807i)T + (1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (1.69 - 0.223i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (0.550 - 0.228i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (4.66 - 2.69i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.654 - 4.97i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (-1.35 - 5.05i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (3.87 + 9.36i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (-1.43 - 2.49i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.83 + 3.71i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (-2.70 + 2.70i)T - 41iT^{2} \) |
| 43 | \( 1 + (-1.10 + 2.65i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (8.38 + 4.83i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.88 + 0.906i)T + (51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (0.0283 + 0.215i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-4.05 - 0.533i)T + (58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (8.41 - 10.9i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-6.41 - 6.41i)T + 71iT^{2} \) |
| 73 | \( 1 + (9.61 + 2.57i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.05 - 2.34i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.11 + 3.77i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (16.1 - 4.31i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + 9.28T + 97T^{2} \) |
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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
−11.99645110464973450099301237585, −11.11614115752226312825131260707, −9.887902962604450548978294114683, −8.826773799504744874919775050136, −8.118026096412298494729224125780, −7.36822263797049273766178208278, −5.57391891074815623380634770767, −4.02637768829147126265110241828, −2.33374936606134708855517106420, −1.56654900866449681244176320733,
2.62158857276584649019378163987, 4.45052355869316055508580935177, 5.02971613096489616235053436469, 6.60100573681581003978105526252, 7.84645374257227040472399176946, 8.761053898497918863056556579881, 9.378033791748534246953257806651, 10.43894966809285565705820090433, 11.09328315545883291476732871249, 13.09321745359764774021278257939
