| L(s) = 1 | + (−1.95 + 1.76i)2-s + (0.650 − 1.46i)3-s + (0.517 − 4.92i)4-s + (0.446 − 2.10i)5-s + (1.30 + 4.01i)6-s + (−1.30 − 2.30i)7-s + (4.56 + 6.28i)8-s + (0.294 + 0.326i)9-s + (2.82 + 4.90i)10-s + (−0.453 + 3.28i)11-s + (−6.85 − 3.95i)12-s + (1.11 − 3.42i)13-s + (6.61 + 2.20i)14-s + (−2.77 − 2.01i)15-s + (−10.3 − 2.20i)16-s + (0.960 − 1.06i)17-s + ⋯ |
| L(s) = 1 | + (−1.38 + 1.24i)2-s + (0.375 − 0.843i)3-s + (0.258 − 2.46i)4-s + (0.199 − 0.939i)5-s + (0.532 + 1.63i)6-s + (−0.493 − 0.869i)7-s + (1.61 + 2.22i)8-s + (0.0981 + 0.108i)9-s + (0.894 + 1.54i)10-s + (−0.136 + 0.990i)11-s + (−1.97 − 1.14i)12-s + (0.309 − 0.951i)13-s + (1.76 + 0.588i)14-s + (−0.717 − 0.521i)15-s + (−2.58 − 0.550i)16-s + (0.233 − 0.258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.551256 - 0.0813842i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.551256 - 0.0813842i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.30 + 2.30i)T \) |
| 11 | \( 1 + (0.453 - 3.28i)T \) |
| good | 2 | \( 1 + (1.95 - 1.76i)T + (0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.650 + 1.46i)T + (-2.00 - 2.22i)T^{2} \) |
| 5 | \( 1 + (-0.446 + 2.10i)T + (-4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (-1.11 + 3.42i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.960 + 1.06i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.173 - 1.65i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (1.02 - 1.77i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.13 - 1.56i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.47 - 6.92i)T + (-28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (0.142 - 0.0632i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-1.86 + 1.35i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 0.0155iT - 43T^{2} \) |
| 47 | \( 1 + (12.7 - 1.34i)T + (45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-6.61 + 1.40i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (1.48 + 0.155i)T + (57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-10.1 - 2.16i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-4.58 - 7.93i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.04 + 12.4i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.327 - 3.12i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-4.55 + 4.09i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-2.94 - 9.05i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-2.07 - 1.19i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.45 + 1.77i)T + (78.4 + 57.0i)T^{2} \) |
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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
−14.64694828601711327397130506248, −13.49907254978486099831351118586, −12.61047015571427397263426168987, −10.47631637528738270322763799645, −9.669335293831892203315830758780, −8.421640222850472599197792577115, −7.58756123383082903612638637946, −6.73005007479554017314198758282, −5.16256631706163598736068538154, −1.28050395726662745365377349817,
2.60074530664321445480924817502, 3.69725068117543752619925798054, 6.55900198328495401800441693606, 8.327737062427088657791659762017, 9.255118120620677885383679951361, 9.979663861815017830662308462815, 10.97837895145436088877084300317, 11.81611182067868436601360468759, 13.16057228958293500330226009447, 14.65874052834780226316233529908
