| L(s) = 1 | − 3-s + 4-s − 2·9-s − 12-s − 3·16-s + 5·27-s + 5·31-s − 2·36-s − 6·37-s + 3·48-s + 4·49-s + 15·53-s − 7·64-s − 9·67-s − 15·71-s + 81-s − 15·89-s − 5·93-s + 6·97-s + 3·103-s + 5·108-s + 6·111-s − 11·121-s + 5·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 2/3·9-s − 0.288·12-s − 3/4·16-s + 0.962·27-s + 0.898·31-s − 1/3·36-s − 0.986·37-s + 0.433·48-s + 4/7·49-s + 2.06·53-s − 7/8·64-s − 1.09·67-s − 1.78·71-s + 1/9·81-s − 1.58·89-s − 0.518·93-s + 0.609·97-s + 0.295·103-s + 0.481·108-s + 0.569·111-s − 121-s + 0.449·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 91 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.138641774942249567494218686022, −7.53258059126941734857294977471, −7.18724875958368574351346895756, −6.69679809007542057142536961357, −6.35662915666782164579168470253, −5.80112125357453558344195791990, −5.48485015186966660312866520559, −4.92209969516231962309625978337, −4.39182612394845217583441195266, −3.89226962219140458079806836181, −3.07413865425757737195029516831, −2.67590504465410219362787567140, −2.03324398628771102650193833768, −1.12599405818960778465998620701, 0,
1.12599405818960778465998620701, 2.03324398628771102650193833768, 2.67590504465410219362787567140, 3.07413865425757737195029516831, 3.89226962219140458079806836181, 4.39182612394845217583441195266, 4.92209969516231962309625978337, 5.48485015186966660312866520559, 5.80112125357453558344195791990, 6.35662915666782164579168470253, 6.69679809007542057142536961357, 7.18724875958368574351346895756, 7.53258059126941734857294977471, 8.138641774942249567494218686022
