| L(s) = 1 | − 2·4-s − 5·7-s − 3·9-s + 2·13-s + 5·25-s + 10·28-s − 15·31-s + 6·36-s + 11·37-s + 8·43-s + 18·49-s − 4·52-s + 15·63-s + 8·64-s + 22·67-s − 24·73-s + 13·79-s + 9·81-s − 10·91-s + 9·97-s − 10·100-s + 6·103-s + 2·109-s − 6·117-s + 22·121-s + 30·124-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s − 1.88·7-s − 9-s + 0.554·13-s + 25-s + 1.88·28-s − 2.69·31-s + 36-s + 1.80·37-s + 1.21·43-s + 18/7·49-s − 0.554·52-s + 1.88·63-s + 64-s + 2.68·67-s − 2.80·73-s + 1.46·79-s + 81-s − 1.04·91-s + 0.913·97-s − 100-s + 0.591·103-s + 0.191·109-s − 0.554·117-s + 2·121-s + 2.69·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5795622880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5795622880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.47108013941927306407568050761, −11.42376862483112469163445446794, −11.35762938656492406484639141040, −10.65370307297799945834805003725, −10.23319924006779617837292489113, −9.514507975512239538876784958308, −9.311893064436845493695050786215, −8.886356997181741808525570788830, −8.604247727463603858204972038708, −7.74223007997395040480041342616, −7.27174023488180175545848110315, −6.55919735561680489925539031100, −6.15648242665337708955878744780, −5.60374273128433385286993151283, −5.11600618082989964288531664496, −4.16755189048506287075935004495, −3.71976796117377401600108823055, −3.10498797724630801994885533464, −2.38166414081353019223230660418, −0.58208706017827288621124243378,
0.58208706017827288621124243378, 2.38166414081353019223230660418, 3.10498797724630801994885533464, 3.71976796117377401600108823055, 4.16755189048506287075935004495, 5.11600618082989964288531664496, 5.60374273128433385286993151283, 6.15648242665337708955878744780, 6.55919735561680489925539031100, 7.27174023488180175545848110315, 7.74223007997395040480041342616, 8.604247727463603858204972038708, 8.886356997181741808525570788830, 9.311893064436845493695050786215, 9.514507975512239538876784958308, 10.23319924006779617837292489113, 10.65370307297799945834805003725, 11.35762938656492406484639141040, 11.42376862483112469163445446794, 12.47108013941927306407568050761
