| L(s) = 1 | − 2·9-s − 4·13-s + 12·17-s + 12·29-s − 4·37-s + 12·41-s − 10·49-s + 12·53-s + 4·61-s − 4·73-s − 5·81-s − 12·89-s − 4·97-s + 12·101-s + 4·109-s + 12·113-s + 8·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 1.10·13-s + 2.91·17-s + 2.22·29-s − 0.657·37-s + 1.87·41-s − 1.42·49-s + 1.64·53-s + 0.512·61-s − 0.468·73-s − 5/9·81-s − 1.27·89-s − 0.406·97-s + 1.19·101-s + 0.383·109-s + 1.12·113-s + 0.739·117-s − 2·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.284619131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.284619131\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.36534498208386853525755121224, −9.703656252458157439228300963800, −9.493169657561115216422783286207, −8.414042470938792927705837581638, −8.407676785947288187996164912905, −7.47121424818625422471458115292, −7.38409000222510611898226618037, −6.43307814142801241380910059979, −5.85740317846722582719715899176, −5.30602566274475700528497955431, −4.82118726362395324834121081499, −3.90074996505307731038192172560, −3.07875775618643284846845606551, −2.58321256178540655780606724063, −1.10363850080882238424508077779,
1.10363850080882238424508077779, 2.58321256178540655780606724063, 3.07875775618643284846845606551, 3.90074996505307731038192172560, 4.82118726362395324834121081499, 5.30602566274475700528497955431, 5.85740317846722582719715899176, 6.43307814142801241380910059979, 7.38409000222510611898226618037, 7.47121424818625422471458115292, 8.407676785947288187996164912905, 8.414042470938792927705837581638, 9.493169657561115216422783286207, 9.703656252458157439228300963800, 10.36534498208386853525755121224
