L(s) = 1 | + 14·13-s + 10·25-s − 2·37-s + 11·49-s − 26·61-s − 34·73-s − 10·97-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 121·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 3.88·13-s + 2·25-s − 0.328·37-s + 11/7·49-s − 3.32·61-s − 3.97·73-s − 1.01·97-s − 0.383·109-s − 2·121-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 + 9.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.145274172\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.145274172\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−11.21201751875047913058671935927, −10.85264818012528431013622729626, −10.52233958930717249313894169368, −10.41623439274679055973854166688, −9.338906886207236034516955166648, −9.005275133993387819880779534966, −8.599536346141706453549999928472, −8.490842121873430371566472986357, −7.74239137126834312935054998100, −7.20217671530449479560334293127, −6.55213855881737620069911805785, −6.21899907388623648826181652843, −5.80810051314631591306814952085, −5.27988116682589768936923727974, −4.27641933862465300523732268190, −4.14149037091446659297479417917, −3.19359405981306094723759491957, −3.01760839698715801962750927644, −1.55995293248542508477899289659, −1.13293753812297903734402838270,
1.13293753812297903734402838270, 1.55995293248542508477899289659, 3.01760839698715801962750927644, 3.19359405981306094723759491957, 4.14149037091446659297479417917, 4.27641933862465300523732268190, 5.27988116682589768936923727974, 5.80810051314631591306814952085, 6.21899907388623648826181652843, 6.55213855881737620069911805785, 7.20217671530449479560334293127, 7.74239137126834312935054998100, 8.490842121873430371566472986357, 8.599536346141706453549999928472, 9.005275133993387819880779534966, 9.338906886207236034516955166648, 10.41623439274679055973854166688, 10.52233958930717249313894169368, 10.85264818012528431013622729626, 11.21201751875047913058671935927