| L(s) = 1 | + 2·5-s − 6·9-s − 4·13-s + 4·17-s + 3·25-s − 4·29-s + 12·37-s − 12·41-s − 12·45-s + 2·49-s + 12·53-s − 4·61-s − 8·65-s − 12·73-s + 27·81-s + 8·85-s − 12·89-s − 28·97-s + 12·101-s + 28·109-s + 36·113-s + 24·117-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s − 1.10·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s + 1.97·37-s − 1.87·41-s − 1.78·45-s + 2/7·49-s + 1.64·53-s − 0.512·61-s − 0.992·65-s − 1.40·73-s + 3·81-s + 0.867·85-s − 1.27·89-s − 2.84·97-s + 1.19·101-s + 2.68·109-s + 3.38·113-s + 2.21·117-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7492222454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7492222454\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−12.70951559228550877361223816630, −12.13050487476073915971902909987, −11.52426214006865205005559601963, −11.11321493608157697547696196545, −10.12246838996141367241613997715, −9.851840183892789826086741161768, −9.022035638415746178025932660186, −8.521590142691914902934262227672, −7.75327442992794184642667314086, −6.97238544391138654785050907556, −5.84906736542976130410760055690, −5.70093888866808737514223874840, −4.76905661119552301505559653794, −3.26180587408034592610487247010, −2.39961978181641342090490014624,
2.39961978181641342090490014624, 3.26180587408034592610487247010, 4.76905661119552301505559653794, 5.70093888866808737514223874840, 5.84906736542976130410760055690, 6.97238544391138654785050907556, 7.75327442992794184642667314086, 8.521590142691914902934262227672, 9.022035638415746178025932660186, 9.851840183892789826086741161768, 10.12246838996141367241613997715, 11.11321493608157697547696196545, 11.52426214006865205005559601963, 12.13050487476073915971902909987, 12.70951559228550877361223816630
