L(s) = 1 | + 4-s − 2·5-s − 8·7-s + 9-s + 4·13-s + 16-s − 2·20-s + 3·25-s − 8·28-s − 6·29-s + 16·35-s + 36-s − 2·45-s + 34·49-s + 4·52-s − 12·53-s − 8·63-s + 64-s − 8·65-s − 8·67-s − 2·80-s + 81-s + 24·83-s − 32·91-s + 3·100-s − 8·103-s − 24·107-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 0.894·5-s − 3.02·7-s + 1/3·9-s + 1.10·13-s + 1/4·16-s − 0.447·20-s + 3/5·25-s − 1.51·28-s − 1.11·29-s + 2.70·35-s + 1/6·36-s − 0.298·45-s + 34/7·49-s + 0.554·52-s − 1.64·53-s − 1.00·63-s + 1/8·64-s − 0.992·65-s − 0.977·67-s − 0.223·80-s + 1/9·81-s + 2.63·83-s − 3.35·91-s + 3/10·100-s − 0.788·103-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6954635271\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6954635271\) |
\(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.990195365669600444284768326297, −7.941231322257043910223245152113, −7.29644524990161901295283348408, −6.76157465626149990699235398037, −6.42217617652666865799421983967, −6.40665814912951379035720239588, −5.66400815492473587620104158041, −5.24277224944307385832356983224, −4.20524949259167664816292353580, −3.94464284360601455806227315490, −3.36585804145949552210873743484, −3.15618781254825975976157473564, −2.56002992860748681371742510547, −1.50318982746469669221964062414, −0.40448336256994679806460088936,
0.40448336256994679806460088936, 1.50318982746469669221964062414, 2.56002992860748681371742510547, 3.15618781254825975976157473564, 3.36585804145949552210873743484, 3.94464284360601455806227315490, 4.20524949259167664816292353580, 5.24277224944307385832356983224, 5.66400815492473587620104158041, 6.40665814912951379035720239588, 6.42217617652666865799421983967, 6.76157465626149990699235398037, 7.29644524990161901295283348408, 7.941231322257043910223245152113, 7.990195365669600444284768326297