| L(s) = 1 | − 9-s + 8·11-s + 8·19-s + 4·29-s − 8·31-s + 4·41-s − 2·49-s + 8·59-s − 12·61-s + 32·71-s + 8·79-s + 81-s − 20·89-s − 8·99-s + 12·101-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.41·11-s + 1.83·19-s + 0.742·29-s − 1.43·31-s + 0.624·41-s − 2/7·49-s + 1.04·59-s − 1.53·61-s + 3.79·71-s + 0.900·79-s + 1/9·81-s − 2.11·89-s − 0.804·99-s + 1.19·101-s + 2.68·109-s + 2.36·121-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 + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.391120043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.391120043\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.330488391983104161778164168589, −8.265595995528917876287035575464, −7.66412074848188367680820773233, −7.37784477894771144501749447914, −6.88369460417687060647464891100, −6.77086699595135258450263919442, −6.34781079241062403551111301118, −5.93590539549645645863818444537, −5.49678042710000296018544707107, −5.31547676740227850415982513295, −4.68658356235272457345085740745, −4.35083624161163919962925768335, −3.91864028512330890369043211768, −3.44153367426976733450502456089, −3.31409451590162164628670898524, −2.75605359372617745715172798664, −1.88897109439793894212810085101, −1.80690725447072681773555992056, −0.907186975100964522292272236971, −0.75643389249367705915271138896,
0.75643389249367705915271138896, 0.907186975100964522292272236971, 1.80690725447072681773555992056, 1.88897109439793894212810085101, 2.75605359372617745715172798664, 3.31409451590162164628670898524, 3.44153367426976733450502456089, 3.91864028512330890369043211768, 4.35083624161163919962925768335, 4.68658356235272457345085740745, 5.31547676740227850415982513295, 5.49678042710000296018544707107, 5.93590539549645645863818444537, 6.34781079241062403551111301118, 6.77086699595135258450263919442, 6.88369460417687060647464891100, 7.37784477894771144501749447914, 7.66412074848188367680820773233, 8.265595995528917876287035575464, 8.330488391983104161778164168589
