| L(s) = 1 | + (−0.996 + 2.33i)2-s + (0.808 + 0.146i)3-s + (−3.05 − 3.19i)4-s + (−1.11 + 3.44i)5-s + (−1.14 + 1.73i)6-s + (1.50 − 1.31i)7-s + (5.75 − 2.15i)8-s + (−2.17 − 0.816i)9-s + (−6.91 − 6.03i)10-s + (2.16 − 0.597i)11-s + (−2.00 − 3.03i)12-s + (5.25 + 1.45i)13-s + (1.56 + 4.80i)14-s + (−1.41 + 2.62i)15-s + (−0.301 + 6.71i)16-s + (−1.37 + 1.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.704 + 1.64i)2-s + (0.466 + 0.0847i)3-s + (−1.52 − 1.59i)4-s + (−0.500 + 1.54i)5-s + (−0.468 + 0.709i)6-s + (0.567 − 0.496i)7-s + (2.03 − 0.763i)8-s + (−0.725 − 0.272i)9-s + (−2.18 − 1.90i)10-s + (0.652 − 0.180i)11-s + (−0.578 − 0.876i)12-s + (1.45 + 0.402i)13-s + (0.417 + 1.28i)14-s + (−0.364 + 0.676i)15-s + (−0.0754 + 1.67i)16-s + (−0.334 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.222802 + 0.650785i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.222802 + 0.650785i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (-8.02 + 2.58i)T \) |
| good | 2 | \( 1 + (0.996 - 2.33i)T + (-1.38 - 1.44i)T^{2} \) |
| 3 | \( 1 + (-0.808 - 0.146i)T + (2.80 + 1.05i)T^{2} \) |
| 5 | \( 1 + (1.11 - 3.44i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-1.50 + 1.31i)T + (0.939 - 6.93i)T^{2} \) |
| 11 | \( 1 + (-2.16 + 0.597i)T + (9.44 - 5.64i)T^{2} \) |
| 13 | \( 1 + (-5.25 - 1.45i)T + (11.1 + 6.66i)T^{2} \) |
| 17 | \( 1 + (1.37 - 1.00i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.64 - 3.05i)T + (-10.4 + 15.8i)T^{2} \) |
| 23 | \( 1 + (-1.37 + 6.04i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (0.550 + 4.06i)T + (-27.9 + 7.71i)T^{2} \) |
| 31 | \( 1 + (-0.159 - 3.56i)T + (-30.8 + 2.77i)T^{2} \) |
| 37 | \( 1 + (1.97 + 8.67i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (4.28 + 5.37i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (3.19 + 0.287i)T + (42.3 + 7.67i)T^{2} \) |
| 47 | \( 1 + (5.09 - 0.924i)T + (44.0 - 16.5i)T^{2} \) |
| 53 | \( 1 + (-0.321 + 0.335i)T + (-2.37 - 52.9i)T^{2} \) |
| 59 | \( 1 + (-4.74 - 7.18i)T + (-23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (1.96 + 1.71i)T + (8.18 + 60.4i)T^{2} \) |
| 67 | \( 1 + (-3.07 - 3.21i)T + (-3.00 + 66.9i)T^{2} \) |
| 73 | \( 1 + (-0.333 + 0.779i)T + (-50.4 - 52.7i)T^{2} \) |
| 79 | \( 1 + (2.75 - 1.03i)T + (59.4 - 51.9i)T^{2} \) |
| 83 | \( 1 + (4.19 + 6.34i)T + (-32.6 + 76.3i)T^{2} \) |
| 89 | \( 1 + (9.63 - 10.0i)T + (-3.99 - 88.9i)T^{2} \) |
| 97 | \( 1 + (-9.09 + 11.4i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−15.07058163657205507450800929765, −14.30149852410709452938267888466, −13.95445195538452627836304488460, −11.40341380966338053233025326903, −10.43705715314499716839550587298, −8.899104597765616105827213646930, −8.081020095075392383976694058256, −6.91498215503052731899151743976, −6.08087196596655846479326998173, −3.81515232127771226077433115585,
1.45024294668587798918381003040, 3.46698595672672174246993311378, 5.02482693311175095641725978951, 8.211263053568373830174272646304, 8.643393465543244672391578386874, 9.495865327437268352258863592012, 11.37741314060731690727952606180, 11.61930252688121605629040614291, 12.95733426607769602984013813202, 13.61356893215205670748364774049
