L(s) = 1 | − 8·17-s + 16·29-s − 8·37-s − 16·41-s − 8·49-s − 16·53-s + 12·61-s − 8·73-s − 9·81-s + 4·89-s + 8·97-s + 16·101-s − 20·109-s + 32·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 1.94·17-s + 2.97·29-s − 1.31·37-s − 2.49·41-s − 8/7·49-s − 2.19·53-s + 1.53·61-s − 0.936·73-s − 81-s + 0.423·89-s + 0.812·97-s + 1.59·101-s − 1.91·109-s + 3.01·113-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7520697990\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7520697990\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−8.365555142726693023650831452123, −7.938712579674749341027435627717, −7.42282764113718541422330001012, −7.04640742334732456526016431617, −6.71136399842423317635684107078, −6.56536775431522621880388433485, −6.14146914088126881397506072413, −5.91642868261680090152416654750, −5.05492272558939153819359159147, −4.90106940327372207356123000447, −4.78501527214180156637575818533, −4.35982032707162211369978842664, −3.61287328949256575702433709953, −3.56119392165859084201390689793, −2.86439867737768881148439064396, −2.65592909961908986369534758015, −1.99604304183357913283386804354, −1.66680322850065496209100505128, −1.08127268940309701371413169651, −0.22620076627417607289201011686,
0.22620076627417607289201011686, 1.08127268940309701371413169651, 1.66680322850065496209100505128, 1.99604304183357913283386804354, 2.65592909961908986369534758015, 2.86439867737768881148439064396, 3.56119392165859084201390689793, 3.61287328949256575702433709953, 4.35982032707162211369978842664, 4.78501527214180156637575818533, 4.90106940327372207356123000447, 5.05492272558939153819359159147, 5.91642868261680090152416654750, 6.14146914088126881397506072413, 6.56536775431522621880388433485, 6.71136399842423317635684107078, 7.04640742334732456526016431617, 7.42282764113718541422330001012, 7.938712579674749341027435627717, 8.365555142726693023650831452123