L(s) = 1 | + 2-s − 4-s + 4·7-s − 3·8-s − 2·9-s − 11-s + 4·14-s − 16-s − 2·18-s − 22-s − 8·23-s − 6·25-s − 4·28-s + 4·29-s + 5·32-s + 2·36-s + 44-s − 8·46-s + 9·49-s − 6·50-s − 8·53-s − 12·56-s + 4·58-s − 8·63-s + 7·64-s + 20·67-s + 12·71-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.51·7-s − 1.06·8-s − 2/3·9-s − 0.301·11-s + 1.06·14-s − 1/4·16-s − 0.471·18-s − 0.213·22-s − 1.66·23-s − 6/5·25-s − 0.755·28-s + 0.742·29-s + 0.883·32-s + 1/3·36-s + 0.150·44-s − 1.17·46-s + 9/7·49-s − 0.848·50-s − 1.09·53-s − 1.60·56-s + 0.525·58-s − 1.00·63-s + 7/8·64-s + 2.44·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9950697187\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9950697187\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−12.27021067188370396709985028906, −11.99297690089808119582251325551, −11.33696894235438898578899248708, −10.89556905826929887808213229599, −9.955763757492334327628645561266, −9.520825901634028793940380124781, −8.509739114195429978253782788936, −8.214170704131043524266746631207, −7.70358982130509525195216328506, −6.48020125855069284363283598797, −5.74125687391337059330576407197, −5.15684451224741078245187981572, −4.43684984520239592288748109335, −3.62979773637590352656751406779, −2.24772405847854638205107848265,
2.24772405847854638205107848265, 3.62979773637590352656751406779, 4.43684984520239592288748109335, 5.15684451224741078245187981572, 5.74125687391337059330576407197, 6.48020125855069284363283598797, 7.70358982130509525195216328506, 8.214170704131043524266746631207, 8.509739114195429978253782788936, 9.520825901634028793940380124781, 9.955763757492334327628645561266, 10.89556905826929887808213229599, 11.33696894235438898578899248708, 11.99297690089808119582251325551, 12.27021067188370396709985028906