| L(s) = 1 | − 2-s − 9·3-s − 31·4-s + 9·6-s + 26·7-s + 63·8-s + 81·9-s + 121·11-s + 279·12-s + 692·13-s − 26·14-s + 929·16-s + 1.44e3·17-s − 81·18-s + 2.16e3·19-s − 234·21-s − 121·22-s + 1.58e3·23-s − 567·24-s − 692·26-s − 729·27-s − 806·28-s − 5.52e3·29-s + 4.79e3·31-s − 2.94e3·32-s − 1.08e3·33-s − 1.44e3·34-s + ⋯ |
| L(s) = 1 | − 0.176·2-s − 0.577·3-s − 0.968·4-s + 0.102·6-s + 0.200·7-s + 0.348·8-s + 1/3·9-s + 0.301·11-s + 0.559·12-s + 1.13·13-s − 0.0354·14-s + 0.907·16-s + 1.21·17-s − 0.0589·18-s + 1.37·19-s − 0.115·21-s − 0.0533·22-s + 0.623·23-s − 0.200·24-s − 0.200·26-s − 0.192·27-s − 0.194·28-s − 1.22·29-s + 0.895·31-s − 0.508·32-s − 0.174·33-s − 0.213·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.683100343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683100343\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + T + p^{5} T^{2} \) |
| 7 | \( 1 - 26 T + p^{5} T^{2} \) |
| 13 | \( 1 - 692 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1442 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2160 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1582 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5526 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4792 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10194 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10622 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8580 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2362 T + p^{5} T^{2} \) |
| 53 | \( 1 - 30804 T + p^{5} T^{2} \) |
| 59 | \( 1 - 6416 T + p^{5} T^{2} \) |
| 61 | \( 1 - 42096 T + p^{5} T^{2} \) |
| 67 | \( 1 - 28444 T + p^{5} T^{2} \) |
| 71 | \( 1 - 45690 T + p^{5} T^{2} \) |
| 73 | \( 1 - 18374 T + p^{5} T^{2} \) |
| 79 | \( 1 + 105214 T + p^{5} T^{2} \) |
| 83 | \( 1 + 62292 T + p^{5} T^{2} \) |
| 89 | \( 1 + 72246 T + p^{5} T^{2} \) |
| 97 | \( 1 + 79262 T + p^{5} T^{2} \) |
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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
−9.637037149264247636351291115680, −8.616966704491774988954526416980, −7.907877422897988887349791506151, −6.92414843572692959947377869018, −5.71534477762410219559408650828, −5.21591246179224001958036505906, −4.07719034094467101873275901610, −3.26948481617356644006910502725, −1.36591222213933405439576121867, −0.73600231103586127416061771542,
0.73600231103586127416061771542, 1.36591222213933405439576121867, 3.26948481617356644006910502725, 4.07719034094467101873275901610, 5.21591246179224001958036505906, 5.71534477762410219559408650828, 6.92414843572692959947377869018, 7.907877422897988887349791506151, 8.616966704491774988954526416980, 9.637037149264247636351291115680
