| L(s) = 1 | − 2·2-s − 4-s + 8·8-s − 8·11-s − 7·16-s + 16·22-s − 16·23-s − 10·25-s − 4·29-s − 14·32-s − 12·37-s − 24·43-s + 8·44-s + 32·46-s + 20·50-s + 20·53-s + 8·58-s + 35·64-s + 8·67-s − 32·71-s + 24·74-s + 16·79-s + 48·86-s − 64·88-s + 16·92-s + 10·100-s − 40·106-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 2.82·8-s − 2.41·11-s − 7/4·16-s + 3.41·22-s − 3.33·23-s − 2·25-s − 0.742·29-s − 2.47·32-s − 1.97·37-s − 3.65·43-s + 1.20·44-s + 4.71·46-s + 2.82·50-s + 2.74·53-s + 1.05·58-s + 35/8·64-s + 0.977·67-s − 3.79·71-s + 2.78·74-s + 1.80·79-s + 5.17·86-s − 6.82·88-s + 1.66·92-s + 100-s − 3.88·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−13.81565191747960206048779386724, −13.41510321545555331973870646050, −13.41510321545555331973870646050, −12.40737222252570732536266891213, −12.40737222252570732536266891213, −11.41526235493003329986136581474, −11.41526235493003329986136581474, −10.22557678234009295397637516324, −10.22557678234009295397637516324, −10.00216059812775063040598752212, −10.00216059812775063040598752212, −8.772288861366294277115033481020, −8.772288861366294277115033481020, −8.063480898156861953425973667312, −8.063480898156861953425973667312, −7.31504642298749821815076432316, −7.31504642298749821815076432316, −5.85934501838508108877594225367, −5.85934501838508108877594225367, −4.87914944321891573254016142709, −4.87914944321891573254016142709, −3.69498409096789705984132005254, −3.69498409096789705984132005254, −1.99068740018288862581437794517, −1.99068740018288862581437794517, 0, 0,
1.99068740018288862581437794517, 1.99068740018288862581437794517, 3.69498409096789705984132005254, 3.69498409096789705984132005254, 4.87914944321891573254016142709, 4.87914944321891573254016142709, 5.85934501838508108877594225367, 5.85934501838508108877594225367, 7.31504642298749821815076432316, 7.31504642298749821815076432316, 8.063480898156861953425973667312, 8.063480898156861953425973667312, 8.772288861366294277115033481020, 8.772288861366294277115033481020, 10.00216059812775063040598752212, 10.00216059812775063040598752212, 10.22557678234009295397637516324, 10.22557678234009295397637516324, 11.41526235493003329986136581474, 11.41526235493003329986136581474, 12.40737222252570732536266891213, 12.40737222252570732536266891213, 13.41510321545555331973870646050, 13.41510321545555331973870646050, 13.81565191747960206048779386724
