| L(s) = 1 | + 4·7-s + 9-s + 4·23-s − 25-s + 4·41-s + 12·47-s − 2·49-s + 4·63-s + 8·71-s − 4·73-s + 81-s + 4·89-s − 4·97-s + 12·103-s − 8·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·161-s + 163-s + 167-s + 6·169-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1/3·9-s + 0.834·23-s − 1/5·25-s + 0.624·41-s + 1.75·47-s − 2/7·49-s + 0.503·63-s + 0.949·71-s − 0.468·73-s + 1/9·81-s + 0.423·89-s − 0.406·97-s + 1.18·103-s − 0.752·113-s − 0.909·121-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 + 1.26·161-s + 0.0783·163-s + 0.0773·167-s + 6/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.221587558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.221587558\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.010491145617605083476351870773, −8.460687659062073508331132456542, −8.061548317513178034733944314162, −7.59291462867186493700473932851, −7.24489141892793086305399814747, −6.65388597794988806319202028461, −6.07248787764964529117609675479, −5.45297028291658394314709952790, −5.03796141753130864842855351444, −4.51416044297849093294124555918, −4.06028195628447761837367673348, −3.30054061241319217567250675863, −2.48949295169962412612476552574, −1.80359649052988157033946617775, −1.00760419767862181709979731337,
1.00760419767862181709979731337, 1.80359649052988157033946617775, 2.48949295169962412612476552574, 3.30054061241319217567250675863, 4.06028195628447761837367673348, 4.51416044297849093294124555918, 5.03796141753130864842855351444, 5.45297028291658394314709952790, 6.07248787764964529117609675479, 6.65388597794988806319202028461, 7.24489141892793086305399814747, 7.59291462867186493700473932851, 8.061548317513178034733944314162, 8.460687659062073508331132456542, 9.010491145617605083476351870773
