| L(s) = 1 | + (−1.32 − 0.496i)2-s + (−1.09 + 0.397i)3-s + (1.50 + 1.31i)4-s + (0.615 − 3.49i)5-s + (1.64 + 0.0159i)6-s + (3.53 − 2.04i)7-s + (−1.34 − 2.48i)8-s + (−1.26 + 1.06i)9-s + (−2.55 + 4.31i)10-s + (−0.260 − 0.150i)11-s + (−2.16 − 0.836i)12-s + (0.546 − 1.50i)13-s + (−5.69 + 0.947i)14-s + (0.715 + 4.05i)15-s + (0.540 + 3.96i)16-s + (3.58 + 3.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.936 − 0.351i)2-s + (−0.629 + 0.229i)3-s + (0.753 + 0.657i)4-s + (0.275 − 1.56i)5-s + (0.670 + 0.00653i)6-s + (1.33 − 0.771i)7-s + (−0.474 − 0.880i)8-s + (−0.421 + 0.354i)9-s + (−0.806 + 1.36i)10-s + (−0.0784 − 0.0452i)11-s + (−0.625 − 0.241i)12-s + (0.151 − 0.416i)13-s + (−1.52 + 0.253i)14-s + (0.184 + 1.04i)15-s + (0.135 + 0.990i)16-s + (0.869 + 0.729i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.511866 - 0.303283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.511866 - 0.303283i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 + 0.496i)T \) |
| 19 | \( 1 + (3.34 - 2.78i)T \) |
| good | 3 | \( 1 + (1.09 - 0.397i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.615 + 3.49i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-3.53 + 2.04i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.260 + 0.150i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.546 + 1.50i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.58 - 3.00i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (3.07 - 0.541i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.344 - 0.410i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.08 - 3.60i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.42iT - 37T^{2} \) |
| 41 | \( 1 + (-2.70 - 7.42i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.23 - 0.393i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.11 - 2.51i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (5.12 - 0.903i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-4.17 - 3.50i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.594 + 3.37i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.29 + 4.44i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.743 - 4.21i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (8.50 - 3.09i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-0.187 + 0.0681i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (12.9 - 7.44i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.26 + 6.21i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.90 + 4.64i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−14.30514922416806513128040798620, −12.89179944170411528796060316313, −11.94215915113691320017959555952, −10.90688517092860690260222984116, −10.02037371520602501313922208579, −8.425853906043312313460920807770, −7.988543549982668123797875393846, −5.77541583861468224658877877713, −4.44300370383208330437670396051, −1.36571741206083680434576937405,
2.39613538419653416340175997019, 5.53602199187925282224681253543, 6.53003223251182967033548964078, 7.67029633149904204601833986581, 8.999695763622572520106344010473, 10.42056648546200346106191870492, 11.31542902508259953089720078976, 11.87357530773972502650264255315, 14.25665140647826707911869925829, 14.67085841562874470140863399213
