| L(s) = 1 | − 9·7-s + 5·13-s − 5·25-s + 18·31-s + 22·37-s + 18·43-s + 47·49-s + 61-s − 27·67-s + 14·73-s − 9·79-s − 45·91-s − 19·97-s − 27·103-s − 4·109-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 13·169-s + 173-s + ⋯ |
| L(s) = 1 | − 3.40·7-s + 1.38·13-s − 25-s + 3.23·31-s + 3.61·37-s + 2.74·43-s + 47/7·49-s + 0.128·61-s − 3.29·67-s + 1.63·73-s − 1.01·79-s − 4.71·91-s − 1.92·97-s − 2.66·103-s − 0.383·109-s + 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 + 169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.351438044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351438044\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.649304019542622494459145834430, −9.537419897530705178550962949928, −9.284618316416540551441264685151, −8.813207284656869274342659048856, −8.062739278088967632716251718838, −8.035786833222693077962625475958, −7.26871237797887970615923864674, −6.89623611599616865831638585242, −6.31967369512688114451687460779, −6.15784028609827297570002826187, −5.99850095907662745116798684150, −5.60818596065546355953762466386, −4.33629274029684484411649105487, −4.28618846900494596557595763794, −3.82291310433862024014930451539, −3.09383752512518789376061432841, −2.72141177884040881689139644468, −2.62462277610483802661090778208, −1.10906952895749879989538321646, −0.57378189885632896911946201687,
0.57378189885632896911946201687, 1.10906952895749879989538321646, 2.62462277610483802661090778208, 2.72141177884040881689139644468, 3.09383752512518789376061432841, 3.82291310433862024014930451539, 4.28618846900494596557595763794, 4.33629274029684484411649105487, 5.60818596065546355953762466386, 5.99850095907662745116798684150, 6.15784028609827297570002826187, 6.31967369512688114451687460779, 6.89623611599616865831638585242, 7.26871237797887970615923864674, 8.035786833222693077962625475958, 8.062739278088967632716251718838, 8.813207284656869274342659048856, 9.284618316416540551441264685151, 9.537419897530705178550962949928, 9.649304019542622494459145834430
