| L(s) = 1 | + 4·5-s − 7-s − 3·9-s − 12·17-s + 2·25-s − 4·35-s − 4·37-s + 4·41-s − 8·43-s − 12·45-s − 16·47-s + 49-s + 3·63-s − 8·67-s + 32·79-s + 9·81-s + 16·83-s − 48·85-s − 12·89-s + 4·101-s − 20·109-s + 12·119-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s − 9-s − 2.91·17-s + 2/5·25-s − 0.676·35-s − 0.657·37-s + 0.624·41-s − 1.21·43-s − 1.78·45-s − 2.33·47-s + 1/7·49-s + 0.377·63-s − 0.977·67-s + 3.60·79-s + 81-s + 1.75·83-s − 5.20·85-s − 1.27·89-s + 0.398·101-s − 1.91·109-s + 1.10·119-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.096051595987974270124894157352, −8.373309408931680629637484917349, −8.188420417684287628984093773695, −7.32119224555753352111842303305, −6.52917361486339326450126089747, −6.42262084813563967264199691637, −6.17908275353188198229198176353, −5.22316409435338364257345412913, −5.16177791023533130836977920691, −4.33405029272495229169198202320, −3.58097051005902557419651238987, −2.79183800612725741835263061431, −2.16319834122457639545883386464, −1.80766814068560455754339816128, 0,
1.80766814068560455754339816128, 2.16319834122457639545883386464, 2.79183800612725741835263061431, 3.58097051005902557419651238987, 4.33405029272495229169198202320, 5.16177791023533130836977920691, 5.22316409435338364257345412913, 6.17908275353188198229198176353, 6.42262084813563967264199691637, 6.52917361486339326450126089747, 7.32119224555753352111842303305, 8.188420417684287628984093773695, 8.373309408931680629637484917349, 9.096051595987974270124894157352
