L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 6·5-s − 4·6-s + 9-s − 12·10-s − 4·12-s + 12·15-s − 4·16-s + 4·17-s + 2·18-s − 12·20-s + 17·25-s − 2·27-s + 24·30-s − 2·31-s − 8·32-s + 8·34-s + 2·36-s − 6·45-s + 8·48-s + 2·49-s + 34·50-s − 8·51-s − 4·54-s + 10·59-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 2.68·5-s − 1.63·6-s + 1/3·9-s − 3.79·10-s − 1.15·12-s + 3.09·15-s − 16-s + 0.970·17-s + 0.471·18-s − 2.68·20-s + 17/5·25-s − 0.384·27-s + 4.38·30-s − 0.359·31-s − 1.41·32-s + 1.37·34-s + 1/3·36-s − 0.894·45-s + 1.15·48-s + 2/7·49-s + 4.80·50-s − 1.12·51-s − 0.544·54-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 125 T^{2} + p^{2} T^{4} \) |
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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
−7.955620829747967184991305746075, −7.37975871079400632387720726747, −7.27657434700434982018023790859, −6.69540733480985854797842074096, −6.12689316095932961420752378920, −5.65148754991620513655064784047, −5.33196366844987424448281360115, −4.62555738340806829213149040527, −4.49493035764248223994648121715, −3.73487047471079695224307298572, −3.62004589786192469949619733666, −3.12875467287827706433861893012, −2.19731876267553041800621884523, −0.802751325936655719005547051772, 0,
0.802751325936655719005547051772, 2.19731876267553041800621884523, 3.12875467287827706433861893012, 3.62004589786192469949619733666, 3.73487047471079695224307298572, 4.49493035764248223994648121715, 4.62555738340806829213149040527, 5.33196366844987424448281360115, 5.65148754991620513655064784047, 6.12689316095932961420752378920, 6.69540733480985854797842074096, 7.27657434700434982018023790859, 7.37975871079400632387720726747, 7.955620829747967184991305746075