| L(s) = 1 | − 2·3-s + 3·9-s − 14·19-s − 2·25-s − 4·27-s − 10·31-s + 2·37-s + 12·47-s + 28·57-s + 4·75-s + 5·81-s + 12·83-s + 20·93-s − 10·103-s + 10·109-s − 4·111-s + 12·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 24·141-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 3.21·19-s − 2/5·25-s − 0.769·27-s − 1.79·31-s + 0.328·37-s + 1.75·47-s + 3.70·57-s + 0.461·75-s + 5/9·81-s + 1.31·83-s + 2.07·93-s − 0.985·103-s + 0.957·109-s − 0.379·111-s + 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.02·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3922007080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3922007080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 131 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.902163139474265767134623469877, −8.860321443583852514870473314905, −8.643676684946635546260721925987, −7.81086686776276912728020671234, −7.65941586414106322012013962778, −7.25896264744544728573173380803, −6.58557942855436153873345219752, −6.51636443359493650086348376868, −6.07894327797552382782959055461, −5.75873176466957925099709555820, −5.20152652135136692443551745839, −4.92912642974434180776879442945, −4.31441989701701102489394116616, −3.90236294391280880426234556317, −3.85989070971442721545316643974, −2.88204096026491647472749100699, −2.13486140637540303783758700631, −2.03987233430507562144818847079, −1.16280677385862750041075195040, −0.24450732074130181368744724318,
0.24450732074130181368744724318, 1.16280677385862750041075195040, 2.03987233430507562144818847079, 2.13486140637540303783758700631, 2.88204096026491647472749100699, 3.85989070971442721545316643974, 3.90236294391280880426234556317, 4.31441989701701102489394116616, 4.92912642974434180776879442945, 5.20152652135136692443551745839, 5.75873176466957925099709555820, 6.07894327797552382782959055461, 6.51636443359493650086348376868, 6.58557942855436153873345219752, 7.25896264744544728573173380803, 7.65941586414106322012013962778, 7.81086686776276912728020671234, 8.643676684946635546260721925987, 8.860321443583852514870473314905, 8.902163139474265767134623469877
