| L(s) = 1 | + 0.856·2-s − 3-s − 1.26·4-s + 5-s − 0.856·6-s + 1.30·7-s − 2.79·8-s + 9-s + 0.856·10-s + 1.76·11-s + 1.26·12-s + 1.33·13-s + 1.11·14-s − 15-s + 0.136·16-s + 3.88·17-s + 0.856·18-s − 1.88·19-s − 1.26·20-s − 1.30·21-s + 1.51·22-s + 2.79·24-s + 25-s + 1.14·26-s − 27-s − 1.65·28-s + 7.10·29-s + ⋯ |
| L(s) = 1 | + 0.605·2-s − 0.577·3-s − 0.633·4-s + 0.447·5-s − 0.349·6-s + 0.492·7-s − 0.989·8-s + 0.333·9-s + 0.270·10-s + 0.533·11-s + 0.365·12-s + 0.369·13-s + 0.298·14-s − 0.258·15-s + 0.0340·16-s + 0.941·17-s + 0.201·18-s − 0.431·19-s − 0.283·20-s − 0.284·21-s + 0.323·22-s + 0.571·24-s + 0.200·25-s + 0.223·26-s − 0.192·27-s − 0.312·28-s + 1.31·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.237605989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237605989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.856T + 2T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 - 1.76T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 19 | \( 1 + 1.88T + 19T^{2} \) |
| 29 | \( 1 - 7.10T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 + 5.03T + 37T^{2} \) |
| 41 | \( 1 - 9.31T + 41T^{2} \) |
| 43 | \( 1 + 5.34T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 - 8.32T + 53T^{2} \) |
| 59 | \( 1 - 0.850T + 59T^{2} \) |
| 61 | \( 1 - 6.87T + 61T^{2} \) |
| 67 | \( 1 + 5.00T + 67T^{2} \) |
| 71 | \( 1 - 7.19T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 + 8.21T + 79T^{2} \) |
| 83 | \( 1 + 1.53T + 83T^{2} \) |
| 89 | \( 1 + 7.04T + 89T^{2} \) |
| 97 | \( 1 - 2.73T + 97T^{2} \) |
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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
−7.898800240763574137782930887830, −6.89865835880789462708370652292, −6.26766545372904830338938687646, −5.64586863524210541154381329898, −5.06541607627030526386922029588, −4.41266380067592663824137318236, −3.71475285413032313269178349509, −2.86518209361833839373929509558, −1.65033400634953565103840767245, −0.74014814276812910188753488613,
0.74014814276812910188753488613, 1.65033400634953565103840767245, 2.86518209361833839373929509558, 3.71475285413032313269178349509, 4.41266380067592663824137318236, 5.06541607627030526386922029588, 5.64586863524210541154381329898, 6.26766545372904830338938687646, 6.89865835880789462708370652292, 7.898800240763574137782930887830
