L(s) = 1 | − 2·3-s + 4-s − 8·5-s + 3·9-s − 11-s − 2·12-s + 16·15-s + 16-s − 8·20-s − 16·23-s + 38·25-s − 4·27-s + 12·31-s + 2·33-s + 3·36-s − 12·37-s − 44-s − 24·45-s + 12·47-s − 2·48-s + 49-s + 4·53-s + 8·55-s + 16·60-s + 64-s + 8·67-s + 32·69-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/2·4-s − 3.57·5-s + 9-s − 0.301·11-s − 0.577·12-s + 4.13·15-s + 1/4·16-s − 1.78·20-s − 3.33·23-s + 38/5·25-s − 0.769·27-s + 2.15·31-s + 0.348·33-s + 1/2·36-s − 1.97·37-s − 0.150·44-s − 3.57·45-s + 1.75·47-s − 0.288·48-s + 1/7·49-s + 0.549·53-s + 1.07·55-s + 2.06·60-s + 1/8·64-s + 0.977·67-s + 3.85·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−7.56853928475057757956492711754, −7.06196128452260537589494409619, −6.84085179777938586722666479286, −6.30548641956857960470901804158, −5.68884256231510340821014445655, −5.42478346201790760477777896086, −4.50182617970418156245344086250, −4.37406932529796151313535127745, −4.10732223636165202223499092609, −3.64360178930743750856977213770, −3.10627234405241606485567063468, −2.48203239562124190952326585155, −1.44573660276187607476186851929, −0.53083690286209524132902681494, 0,
0.53083690286209524132902681494, 1.44573660276187607476186851929, 2.48203239562124190952326585155, 3.10627234405241606485567063468, 3.64360178930743750856977213770, 4.10732223636165202223499092609, 4.37406932529796151313535127745, 4.50182617970418156245344086250, 5.42478346201790760477777896086, 5.68884256231510340821014445655, 6.30548641956857960470901804158, 6.84085179777938586722666479286, 7.06196128452260537589494409619, 7.56853928475057757956492711754