| L(s) = 1 | + (0.222 + 1.98i)2-s + (0.512 − 1.91i)3-s + (−3.90 + 0.885i)4-s + (−2.85 − 4.10i)5-s + (3.91 + 0.592i)6-s + (5.83 − 3.86i)7-s + (−2.62 − 7.55i)8-s + (4.40 + 2.54i)9-s + (7.52 − 6.59i)10-s + (14.5 − 8.39i)11-s + (−0.305 + 7.91i)12-s + (−2.26 + 2.26i)13-s + (8.97 + 10.7i)14-s + (−9.31 + 3.35i)15-s + (14.4 − 6.90i)16-s + (3.33 − 12.4i)17-s + ⋯ |
| L(s) = 1 | + (0.111 + 0.993i)2-s + (0.170 − 0.637i)3-s + (−0.975 + 0.221i)4-s + (−0.571 − 0.820i)5-s + (0.652 + 0.0987i)6-s + (0.833 − 0.551i)7-s + (−0.328 − 0.944i)8-s + (0.488 + 0.282i)9-s + (0.752 − 0.659i)10-s + (1.32 − 0.763i)11-s + (−0.0254 + 0.659i)12-s + (−0.174 + 0.174i)13-s + (0.641 + 0.767i)14-s + (−0.620 + 0.223i)15-s + (0.902 − 0.431i)16-s + (0.196 − 0.731i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.42594 - 0.224143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42594 - 0.224143i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.222 - 1.98i)T \) |
| 5 | \( 1 + (2.85 + 4.10i)T \) |
| 7 | \( 1 + (-5.83 + 3.86i)T \) |
| good | 3 | \( 1 + (-0.512 + 1.91i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (-14.5 + 8.39i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (2.26 - 2.26i)T - 169iT^{2} \) |
| 17 | \( 1 + (-3.33 + 12.4i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (30.4 + 17.5i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-1.55 - 5.80i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 42.1iT - 841T^{2} \) |
| 31 | \( 1 + (0.982 + 1.70i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (2.93 + 10.9i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 52.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-30.7 + 30.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-7.95 - 29.7i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-0.674 + 2.51i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (62.4 - 36.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-2.55 - 1.47i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-15.2 + 57.0i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-83.3 - 22.3i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (1.12 - 1.95i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-73.7 - 73.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-33.2 + 57.5i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-37.9 - 37.9i)T + 9.40e3iT^{2} \) |
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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
−13.07018042405608593424747761527, −12.16468523841184442120677466705, −10.98286515316560301712181781338, −9.211702246082553837316303208542, −8.447663884887481531195433688675, −7.48260614817454348454886701724, −6.61867528830922306179026462288, −4.94408168684898424814382522519, −4.05063516360996597943142243768, −1.06385977039674240383523410398,
1.99694350685434956971663245869, 3.79996004514951934519607882127, 4.44756653350262996371829798048, 6.28659997619649189213457045430, 7.971500808373716317027486259763, 9.063844287935987189067883692168, 10.13255801038572712252945856845, 10.86947748673389267759149802412, 11.98100557900383486348702598044, 12.50776987572334359175773291670
