| L(s) = 1 | + 2-s + 4-s + 8-s − 6·9-s + 16-s − 12·17-s − 6·18-s + 16·23-s − 6·25-s − 2·31-s + 32-s − 12·34-s − 6·36-s − 12·41-s + 16·46-s − 16·47-s − 14·49-s − 6·50-s − 2·62-s + 64-s − 12·68-s + 16·71-s − 6·72-s + 20·73-s − 16·79-s + 27·81-s − 12·82-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2·9-s + 1/4·16-s − 2.91·17-s − 1.41·18-s + 3.33·23-s − 6/5·25-s − 0.359·31-s + 0.176·32-s − 2.05·34-s − 36-s − 1.87·41-s + 2.35·46-s − 2.33·47-s − 2·49-s − 0.848·50-s − 0.254·62-s + 1/8·64-s − 1.45·68-s + 1.89·71-s − 0.707·72-s + 2.34·73-s − 1.80·79-s + 3·81-s − 1.32·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−9.056498267458590117851162621111, −8.714468530691354459730603161713, −8.307477897920151884753308617841, −7.76857292337562945056886541715, −6.81859577538004307044633877685, −6.64449933506524691025726128618, −6.32858579420662158347624019611, −5.39201619892659079650657036147, −5.00958996845543469784857559007, −4.74203912193695697715499891246, −3.69780406318668437933515679302, −3.16188554299090604324575557253, −2.60552830936859898841087279326, −1.84479362740755814713596587903, 0,
1.84479362740755814713596587903, 2.60552830936859898841087279326, 3.16188554299090604324575557253, 3.69780406318668437933515679302, 4.74203912193695697715499891246, 5.00958996845543469784857559007, 5.39201619892659079650657036147, 6.32858579420662158347624019611, 6.64449933506524691025726128618, 6.81859577538004307044633877685, 7.76857292337562945056886541715, 8.307477897920151884753308617841, 8.714468530691354459730603161713, 9.056498267458590117851162621111
