| L(s) = 1 | − 2·3-s − 2·4-s + 7-s + 9-s + 3·11-s + 4·12-s − 4·13-s + 4·16-s + 3·17-s − 2·21-s + 4·27-s − 2·28-s − 6·29-s + 4·31-s − 6·33-s − 2·36-s + 2·37-s + 8·39-s + 6·41-s + 43-s − 6·44-s + 3·47-s − 8·48-s − 6·49-s − 6·51-s + 8·52-s + 12·53-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.15·12-s − 1.10·13-s + 16-s + 0.727·17-s − 0.436·21-s + 0.769·27-s − 0.377·28-s − 1.11·29-s + 0.718·31-s − 1.04·33-s − 1/3·36-s + 0.328·37-s + 1.28·39-s + 0.937·41-s + 0.152·43-s − 0.904·44-s + 0.437·47-s − 1.15·48-s − 6/7·49-s − 0.840·51-s + 1.10·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8468614901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8468614901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.61870996574027872045561708698, −7.09492209075336668193271147178, −6.10586687521184570813995836925, −5.64391485887755959849356225387, −5.00098609153487198609192007679, −4.41624102780538194892052122814, −3.73413760442576275467138367739, −2.65038758050426083294619198007, −1.36482906238741549229048024476, −0.52679051057606925388940696432,
0.52679051057606925388940696432, 1.36482906238741549229048024476, 2.65038758050426083294619198007, 3.73413760442576275467138367739, 4.41624102780538194892052122814, 5.00098609153487198609192007679, 5.64391485887755959849356225387, 6.10586687521184570813995836925, 7.09492209075336668193271147178, 7.61870996574027872045561708698
