| L(s) = 1 | + (1.05 − 0.945i)2-s + (−0.195 + 0.980i)3-s + (0.211 − 1.98i)4-s + (3.63 + 2.42i)5-s + (0.722 + 1.21i)6-s + (−2.55 + 1.05i)7-s + (−1.65 − 2.29i)8-s + (−0.923 − 0.382i)9-s + (6.11 − 0.882i)10-s + (0.184 − 0.0366i)11-s + (1.90 + 0.595i)12-s + (2.04 − 1.36i)13-s + (−1.68 + 3.53i)14-s + (−3.08 + 3.08i)15-s + (−3.91 − 0.840i)16-s + (−3.38 − 3.38i)17-s + ⋯ |
| L(s) = 1 | + (0.743 − 0.668i)2-s + (−0.112 + 0.566i)3-s + (0.105 − 0.994i)4-s + (1.62 + 1.08i)5-s + (0.294 + 0.496i)6-s + (−0.966 + 0.400i)7-s + (−0.586 − 0.809i)8-s + (−0.307 − 0.127i)9-s + (1.93 − 0.279i)10-s + (0.0556 − 0.0110i)11-s + (0.551 + 0.171i)12-s + (0.567 − 0.379i)13-s + (−0.450 + 0.943i)14-s + (−0.797 + 0.797i)15-s + (−0.977 − 0.210i)16-s + (−0.821 − 0.821i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82061 - 0.200334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82061 - 0.200334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.05 + 0.945i)T \) |
| 3 | \( 1 + (0.195 - 0.980i)T \) |
| good | 5 | \( 1 + (-3.63 - 2.42i)T + (1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (2.55 - 1.05i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.184 + 0.0366i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-2.04 + 1.36i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (3.38 + 3.38i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.72 + 2.57i)T + (-7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-1.69 + 4.10i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (6.98 + 1.38i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 1.32iT - 31T^{2} \) |
| 37 | \( 1 + (0.633 - 0.948i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (1.49 - 3.61i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-2.50 - 12.5i)T + (-39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-1.12 - 1.12i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.47 + 1.28i)T + (48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (10.9 + 7.34i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (-2.31 + 11.6i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (0.0474 - 0.238i)T + (-61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (-11.1 + 4.62i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-5.11 - 2.11i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.69 + 1.69i)T - 79iT^{2} \) |
| 83 | \( 1 + (-3.46 - 5.18i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (0.185 + 0.448i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 2.48iT - 97T^{2} \) |
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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
−12.81034590774337445737851839924, −11.19347083605123598627105262823, −10.71786182502947185588314010201, −9.623776492725962040262617662623, −9.269426349410416840993575443790, −6.62049230885215522971419414286, −6.17173679566039281913477953197, −5.02482688322831913633561899275, −3.26708419203380631343845729739, −2.38722642912428226755812687499,
1.98571574005298247852492220874, 3.94016218587766693607129432601, 5.49168688341988010441766846605, 6.11331134985206981249851381153, 7.05117739471814623426959511490, 8.604843885863043484046872904047, 9.281134467192606510586081025428, 10.61281238815935577969971689320, 12.15974823791630877516122970790, 12.98999668653053912320783083484
