| L(s) = 1 | − 3-s + 9-s + 6·13-s − 4·16-s − 6·17-s − 6·23-s + 25-s − 27-s − 12·29-s − 6·39-s − 2·43-s + 4·48-s − 11·49-s + 6·51-s − 6·53-s − 2·61-s + 6·69-s − 75-s + 10·79-s + 81-s + 12·87-s − 22·103-s + 30·107-s − 12·113-s + 6·117-s + 5·121-s + 127-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.66·13-s − 16-s − 1.45·17-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 2.22·29-s − 0.960·39-s − 0.304·43-s + 0.577·48-s − 1.57·49-s + 0.840·51-s − 0.824·53-s − 0.256·61-s + 0.722·69-s − 0.115·75-s + 1.12·79-s + 1/9·81-s + 1.28·87-s − 2.16·103-s + 2.90·107-s − 1.12·113-s + 0.554·117-s + 5/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
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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
−9.347691401005882567688095596754, −8.764849521648139142047140943795, −8.281556546433321983313347467846, −7.81740187320036846314375827327, −7.09760123551372754998999268465, −6.66653722702651311928037518697, −6.13322868742039769534864861513, −5.86367968665133606569726330162, −5.04544165262948069508341478310, −4.49184588960013616728336196828, −3.90941267180726104206606018176, −3.39462459679807280040118715115, −2.21739806609913236324033597450, −1.61522392743464931323005254175, 0,
1.61522392743464931323005254175, 2.21739806609913236324033597450, 3.39462459679807280040118715115, 3.90941267180726104206606018176, 4.49184588960013616728336196828, 5.04544165262948069508341478310, 5.86367968665133606569726330162, 6.13322868742039769534864861513, 6.66653722702651311928037518697, 7.09760123551372754998999268465, 7.81740187320036846314375827327, 8.281556546433321983313347467846, 8.764849521648139142047140943795, 9.347691401005882567688095596754
