| L(s) = 1 | + 4-s + 2·7-s − 3·9-s + 12·13-s + 16-s + 2·28-s + 16·31-s − 3·36-s + 20·37-s − 8·43-s + 3·49-s + 12·52-s − 28·61-s − 6·63-s + 64-s + 24·67-s − 4·73-s − 16·79-s + 9·81-s + 24·91-s − 4·97-s − 32·103-s + 12·109-s + 2·112-s − 36·117-s − 6·121-s + 16·124-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s − 9-s + 3.32·13-s + 1/4·16-s + 0.377·28-s + 2.87·31-s − 1/2·36-s + 3.28·37-s − 1.21·43-s + 3/7·49-s + 1.66·52-s − 3.58·61-s − 0.755·63-s + 1/8·64-s + 2.93·67-s − 0.468·73-s − 1.80·79-s + 81-s + 2.51·91-s − 0.406·97-s − 3.15·103-s + 1.14·109-s + 0.188·112-s − 3.32·117-s − 0.545·121-s + 1.43·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.417866420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.417866420\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−8.089729535357517933440155073370, −8.027088624732001759148453980283, −7.15015901425524002945866203830, −6.56718920518778472538358027386, −6.17150280411636765429661983697, −6.01430128602422585899125610915, −5.66533618240794239616734860431, −4.84400878985485571871578959566, −4.36823223624486251008730105784, −3.99067474905046324961213028566, −3.16804411819433045396819389047, −2.98711940319117381089884417433, −2.22797785605031286332443349075, −1.27008084899723889248060652436, −1.02699351351041065390998044717,
1.02699351351041065390998044717, 1.27008084899723889248060652436, 2.22797785605031286332443349075, 2.98711940319117381089884417433, 3.16804411819433045396819389047, 3.99067474905046324961213028566, 4.36823223624486251008730105784, 4.84400878985485571871578959566, 5.66533618240794239616734860431, 6.01430128602422585899125610915, 6.17150280411636765429661983697, 6.56718920518778472538358027386, 7.15015901425524002945866203830, 8.027088624732001759148453980283, 8.089729535357517933440155073370
