| L(s) = 1 | + 2·2-s − 4-s + 7-s − 8·8-s + 9-s + 2·14-s − 7·16-s + 2·18-s + 16·23-s + 25-s − 28-s − 4·29-s + 14·32-s − 36-s − 4·37-s + 8·43-s + 32·46-s + 49-s + 2·50-s + 20·53-s − 8·56-s − 8·58-s + 63-s + 35·64-s + 8·67-s − 24·71-s − 8·72-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 0.377·7-s − 2.82·8-s + 1/3·9-s + 0.534·14-s − 7/4·16-s + 0.471·18-s + 3.33·23-s + 1/5·25-s − 0.188·28-s − 0.742·29-s + 2.47·32-s − 1/6·36-s − 0.657·37-s + 1.21·43-s + 4.71·46-s + 1/7·49-s + 0.282·50-s + 2.74·53-s − 1.06·56-s − 1.05·58-s + 0.125·63-s + 35/8·64-s + 0.977·67-s − 2.84·71-s − 0.942·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77175 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77175 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.106996777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106996777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.386500711260631787757553076630, −9.357583073556529155557794013065, −8.843078054411908501657519136744, −8.447489066361171687346291758504, −7.69164818258356184893378968692, −6.97944382340517249810347443111, −6.67009631001529545506185276841, −5.69208011630684437322158261218, −5.37573342552736399524296425305, −5.01564976002989168468776414543, −4.33309058948755389799390879771, −3.94068846921577588199342611811, −3.18167275817234212778601509206, −2.63294756479994443235170715895, −0.982469269817440515290174005672,
0.982469269817440515290174005672, 2.63294756479994443235170715895, 3.18167275817234212778601509206, 3.94068846921577588199342611811, 4.33309058948755389799390879771, 5.01564976002989168468776414543, 5.37573342552736399524296425305, 5.69208011630684437322158261218, 6.67009631001529545506185276841, 6.97944382340517249810347443111, 7.69164818258356184893378968692, 8.447489066361171687346291758504, 8.843078054411908501657519136744, 9.357583073556529155557794013065, 9.386500711260631787757553076630
