| L(s) = 1 | − 2-s − 4-s + 3·8-s − 4·9-s + 6·13-s − 16-s − 4·17-s + 4·18-s − 4·19-s − 4·25-s − 6·26-s − 5·32-s + 4·34-s + 4·36-s + 4·38-s + 12·43-s − 8·47-s − 14·49-s + 4·50-s − 6·52-s − 2·53-s + 12·59-s + 7·64-s − 12·67-s + 4·68-s − 12·72-s + 4·76-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 4/3·9-s + 1.66·13-s − 1/4·16-s − 0.970·17-s + 0.942·18-s − 0.917·19-s − 4/5·25-s − 1.17·26-s − 0.883·32-s + 0.685·34-s + 2/3·36-s + 0.648·38-s + 1.82·43-s − 1.16·47-s − 2·49-s + 0.565·50-s − 0.832·52-s − 0.274·53-s + 1.56·59-s + 7/8·64-s − 1.46·67-s + 0.485·68-s − 1.41·72-s + 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 17 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
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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
−9.489085064039932945941352684032, −8.901869891792090988443503544386, −8.573541937944932621204340410873, −8.244299230435508944325558614124, −7.83728244951620955897515038085, −6.99903976181307509729847467500, −6.39382338444771627597108707867, −5.93973800142688428508748638829, −5.41528180858206820304837242879, −4.58083151074979232965665155560, −4.06204669979065598143458941716, −3.40460075842175198231400425195, −2.46421653499910406522937777179, −1.46507840550827564740297624245, 0,
1.46507840550827564740297624245, 2.46421653499910406522937777179, 3.40460075842175198231400425195, 4.06204669979065598143458941716, 4.58083151074979232965665155560, 5.41528180858206820304837242879, 5.93973800142688428508748638829, 6.39382338444771627597108707867, 6.99903976181307509729847467500, 7.83728244951620955897515038085, 8.244299230435508944325558614124, 8.573541937944932621204340410873, 8.901869891792090988443503544386, 9.489085064039932945941352684032
