| L(s) = 1 | + 4·13-s − 12·23-s + 2·25-s − 4·37-s − 20·47-s + 12·49-s + 24·59-s + 20·61-s − 4·71-s − 20·73-s + 8·83-s + 24·97-s − 16·107-s − 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1.10·13-s − 2.50·23-s + 2/5·25-s − 0.657·37-s − 2.91·47-s + 12/7·49-s + 3.12·59-s + 2.56·61-s − 0.474·71-s − 2.34·73-s + 0.878·83-s + 2.43·97-s − 1.54·107-s − 1.91·109-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 − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.074188180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.074188180\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−8.514279705462625924714808563202, −8.063185903273047506396163996512, −7.973585789856338936828036925021, −7.44099731351353017055810124025, −6.93192915184719534918215991765, −6.75383540289388016043689569726, −6.16362705844462686329400091128, −6.15975675213146004532948389346, −5.43425506182866440690025483124, −5.34930459758865093167425798010, −4.88717081131835176986691911032, −4.11834416624773577878285300205, −3.95453226816894290730954599198, −3.79972161800046962695423851304, −3.10480788238594224027819840609, −2.70818312196148287393452785584, −1.96826483377392857615872328186, −1.86607018937731840724692738260, −1.08652884750568271833373000645, −0.43000508201211822424306888142,
0.43000508201211822424306888142, 1.08652884750568271833373000645, 1.86607018937731840724692738260, 1.96826483377392857615872328186, 2.70818312196148287393452785584, 3.10480788238594224027819840609, 3.79972161800046962695423851304, 3.95453226816894290730954599198, 4.11834416624773577878285300205, 4.88717081131835176986691911032, 5.34930459758865093167425798010, 5.43425506182866440690025483124, 6.15975675213146004532948389346, 6.16362705844462686329400091128, 6.75383540289388016043689569726, 6.93192915184719534918215991765, 7.44099731351353017055810124025, 7.973585789856338936828036925021, 8.063185903273047506396163996512, 8.514279705462625924714808563202
