| L(s) = 1 | + 2-s − 3-s + 4-s − 8·5-s − 6-s + 8-s − 2·9-s − 8·10-s − 12-s + 8·15-s + 16-s − 2·18-s − 2·19-s − 8·20-s − 2·23-s − 24-s + 38·25-s + 5·27-s − 10·29-s + 8·30-s + 32-s − 2·36-s − 2·38-s − 8·40-s + 8·43-s + 16·45-s − 2·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 3.57·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 2.52·10-s − 0.288·12-s + 2.06·15-s + 1/4·16-s − 0.471·18-s − 0.458·19-s − 1.78·20-s − 0.417·23-s − 0.204·24-s + 38/5·25-s + 0.962·27-s − 1.85·29-s + 1.46·30-s + 0.176·32-s − 1/3·36-s − 0.324·38-s − 1.26·40-s + 1.21·43-s + 2.38·45-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.408486909011940238981299566497, −7.69091964872191681066145092749, −7.51365445718420633059750043792, −7.25038064772684706053554512263, −6.52592917100039061776544750020, −6.14974687185144050865472542876, −5.28069074518443602737935578596, −5.05050378173441072659755683035, −4.30490757572779981002207201168, −3.86057015395193111701613139713, −3.81281010152338816670306094022, −3.08766645378616687637023050824, −2.40005554242355398519906349614, −0.807536518397022830754994430552, 0,
0.807536518397022830754994430552, 2.40005554242355398519906349614, 3.08766645378616687637023050824, 3.81281010152338816670306094022, 3.86057015395193111701613139713, 4.30490757572779981002207201168, 5.05050378173441072659755683035, 5.28069074518443602737935578596, 6.14974687185144050865472542876, 6.52592917100039061776544750020, 7.25038064772684706053554512263, 7.51365445718420633059750043792, 7.69091964872191681066145092749, 8.408486909011940238981299566497
