L(s) = 1 | − 2-s + 4-s − 8·7-s − 8-s − 6·9-s + 8·14-s + 16-s − 4·17-s + 6·18-s + 2·23-s + 6·25-s − 8·28-s − 32-s + 4·34-s − 6·36-s + 12·41-s − 2·46-s + 34·49-s − 6·50-s + 8·56-s + 48·63-s + 64-s − 4·68-s + 6·72-s + 12·73-s − 24·79-s + 27·81-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 3.02·7-s − 0.353·8-s − 2·9-s + 2.13·14-s + 1/4·16-s − 0.970·17-s + 1.41·18-s + 0.417·23-s + 6/5·25-s − 1.51·28-s − 0.176·32-s + 0.685·34-s − 36-s + 1.87·41-s − 0.294·46-s + 34/7·49-s − 0.848·50-s + 1.06·56-s + 6.04·63-s + 1/8·64-s − 0.485·68-s + 0.707·72-s + 1.40·73-s − 2.70·79-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3088601721\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3088601721\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.594811427439617650906242640583, −9.375095722993834307181268180399, −8.971935452602447588346290189052, −8.605279085672652507531936126565, −7.958466166331513881205682772342, −7.12318229360480250366687068787, −6.69441913641732873591204403077, −6.36052344432786852577653829766, −5.84432417959727908710252168629, −5.38018932794932277292162659597, −4.20755290673481666320643326040, −3.38951584690182544656335359209, −2.80957070198421100444882721314, −2.63232737474552247848840602025, −0.45732431016122368940787329792,
0.45732431016122368940787329792, 2.63232737474552247848840602025, 2.80957070198421100444882721314, 3.38951584690182544656335359209, 4.20755290673481666320643326040, 5.38018932794932277292162659597, 5.84432417959727908710252168629, 6.36052344432786852577653829766, 6.69441913641732873591204403077, 7.12318229360480250366687068787, 7.958466166331513881205682772342, 8.605279085672652507531936126565, 8.971935452602447588346290189052, 9.375095722993834307181268180399, 9.594811427439617650906242640583