| L(s) = 1 | + 4·11-s + 4·13-s + 4·23-s + 4·25-s + 4·37-s − 12·47-s − 2·49-s + 8·59-s − 4·61-s − 20·71-s + 16·73-s + 28·83-s + 8·97-s − 16·107-s + 12·109-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 16·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 1.10·13-s + 0.834·23-s + 4/5·25-s + 0.657·37-s − 1.75·47-s − 2/7·49-s + 1.04·59-s − 0.512·61-s − 2.37·71-s + 1.87·73-s + 3.07·83-s + 0.812·97-s − 1.54·107-s + 1.14·109-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.33·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192463680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192463680\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 112 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.700329529267379927301840149735, −8.471770036152253569331849101775, −7.83978338420570512858223363668, −7.40866608920881710228865754425, −6.64500992415817695511138673670, −6.55693183899934184786508119538, −6.07261784940020448598726563292, −5.40195356894173547164403640992, −4.83026756570185645145412145147, −4.38741012016571128445016163107, −3.58667290912272838986984704357, −3.40160651343677777478326998554, −2.52226994101748165578857699934, −1.60129198417072837165039976180, −0.944498923343102626899566932708,
0.944498923343102626899566932708, 1.60129198417072837165039976180, 2.52226994101748165578857699934, 3.40160651343677777478326998554, 3.58667290912272838986984704357, 4.38741012016571128445016163107, 4.83026756570185645145412145147, 5.40195356894173547164403640992, 6.07261784940020448598726563292, 6.55693183899934184786508119538, 6.64500992415817695511138673670, 7.40866608920881710228865754425, 7.83978338420570512858223363668, 8.471770036152253569331849101775, 8.700329529267379927301840149735
