| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 2·7-s − 8-s + 9-s + 10-s + 12-s + 2·14-s − 15-s + 16-s − 17-s − 18-s − 6·19-s − 20-s − 2·21-s − 24-s + 25-s + 27-s − 2·28-s + 6·29-s + 30-s + 4·31-s − 32-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.436·21-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.377·28-s + 1.11·29-s + 0.182·30-s + 0.718·31-s − 0.176·32-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.456190736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.456190736\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.03657211877151, −13.32087509654560, −12.82659929094766, −12.55429632723809, −11.87680083099684, −11.43916330354576, −10.74887444169175, −10.34563888955421, −9.964225072191776, −9.280132793782286, −8.893880067630880, −8.406204036723643, −7.985807568765031, −7.401921413961360, −6.755095233593087, −6.483408862237044, −5.870771235860570, −5.042674195098004, −4.291215484996871, −3.912817701730238, −3.174836313535821, −2.542329205001904, −2.168803635912626, −1.123669037683048, −0.4624589667672753,
0.4624589667672753, 1.123669037683048, 2.168803635912626, 2.542329205001904, 3.174836313535821, 3.912817701730238, 4.291215484996871, 5.042674195098004, 5.870771235860570, 6.483408862237044, 6.755095233593087, 7.401921413961360, 7.985807568765031, 8.406204036723643, 8.893880067630880, 9.280132793782286, 9.964225072191776, 10.34563888955421, 10.74887444169175, 11.43916330354576, 11.87680083099684, 12.55429632723809, 12.82659929094766, 13.32087509654560, 14.03657211877151
