L(s) = 1 | + 3·3-s − 7-s + 6·9-s + 9·19-s − 3·21-s + 5·25-s + 9·27-s − 15·31-s − 37-s + 10·43-s − 6·49-s + 27·57-s + 12·61-s − 6·63-s + 11·67-s − 27·73-s + 15·75-s − 13·79-s + 9·81-s − 45·93-s − 33·103-s + 17·109-s − 3·111-s − 11·121-s + 127-s + 30·129-s + 131-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s + 2.06·19-s − 0.654·21-s + 25-s + 1.73·27-s − 2.69·31-s − 0.164·37-s + 1.52·43-s − 6/7·49-s + 3.57·57-s + 1.53·61-s − 0.755·63-s + 1.34·67-s − 3.16·73-s + 1.73·75-s − 1.46·79-s + 81-s − 4.66·93-s − 3.25·103-s + 1.62·109-s − 0.284·111-s − 121-s + 0.0887·127-s + 2.64·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.931350310\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.931350310\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.67660316037631171716828806208, −11.41941785603222522887957765342, −10.69686699709282516646201134262, −10.33981794205294891009798878631, −9.533009708702607669203660333621, −9.531179361467725547520808615724, −9.089326327690703632501222123323, −8.525240011691245440657276104083, −8.107002892703647091153826134274, −7.49347509077663509375304093656, −7.04881314349125167750816694374, −6.96992794580905081562518928567, −5.69065440049857888425647511780, −5.53399340368707844810123254047, −4.59530871416190169385855182024, −3.97616432998908503075031220488, −3.29200566794896011726735625410, −3.04444386945885656669642568983, −2.18030472597969985171433190390, −1.29473696278255747630958387854,
1.29473696278255747630958387854, 2.18030472597969985171433190390, 3.04444386945885656669642568983, 3.29200566794896011726735625410, 3.97616432998908503075031220488, 4.59530871416190169385855182024, 5.53399340368707844810123254047, 5.69065440049857888425647511780, 6.96992794580905081562518928567, 7.04881314349125167750816694374, 7.49347509077663509375304093656, 8.107002892703647091153826134274, 8.525240011691245440657276104083, 9.089326327690703632501222123323, 9.531179361467725547520808615724, 9.533009708702607669203660333621, 10.33981794205294891009798878631, 10.69686699709282516646201134262, 11.41941785603222522887957765342, 11.67660316037631171716828806208