| L(s) = 1 | + 4·9-s + 6·13-s + 6·17-s − 5·25-s + 4·29-s + 10·37-s − 8·41-s − 8·49-s + 10·53-s + 4·61-s − 6·73-s + 7·81-s + 10·97-s + 24·101-s − 12·109-s − 2·113-s + 24·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 1.66·13-s + 1.45·17-s − 25-s + 0.742·29-s + 1.64·37-s − 1.24·41-s − 8/7·49-s + 1.37·53-s + 0.512·61-s − 0.702·73-s + 7/9·81-s + 1.01·97-s + 2.38·101-s − 1.14·109-s − 0.188·113-s + 2.21·117-s − 0.545·121-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.538656265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.538656265\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 124 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−8.732450807243708035321836564143, −8.049565377768392452404490836135, −7.76158266910502995399996162077, −7.38296501641160723175930151805, −6.69687310308520085135677271992, −6.33780124174483103118358740877, −5.89985728132610784467114201075, −5.35964050997371338869439150864, −4.75305729083389703346554128536, −4.18583318270116369277001129362, −3.68101022034315218045289228267, −3.31851051610653643701867541534, −2.39354665428930520330351840367, −1.49571416740470988900861860529, −1.02426979124544176458152180790,
1.02426979124544176458152180790, 1.49571416740470988900861860529, 2.39354665428930520330351840367, 3.31851051610653643701867541534, 3.68101022034315218045289228267, 4.18583318270116369277001129362, 4.75305729083389703346554128536, 5.35964050997371338869439150864, 5.89985728132610784467114201075, 6.33780124174483103118358740877, 6.69687310308520085135677271992, 7.38296501641160723175930151805, 7.76158266910502995399996162077, 8.049565377768392452404490836135, 8.732450807243708035321836564143
