| L(s) = 1 | + 2·3-s + 3·4-s + 3·9-s + 6·12-s − 6·13-s + 5·16-s + 4·17-s − 16·23-s − 25-s + 4·27-s + 4·29-s + 9·36-s − 12·39-s − 8·43-s + 10·48-s + 10·49-s + 8·51-s − 18·52-s − 12·53-s + 20·61-s + 3·64-s + 12·68-s − 32·69-s − 2·75-s − 16·79-s + 5·81-s + 8·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 3/2·4-s + 9-s + 1.73·12-s − 1.66·13-s + 5/4·16-s + 0.970·17-s − 3.33·23-s − 1/5·25-s + 0.769·27-s + 0.742·29-s + 3/2·36-s − 1.92·39-s − 1.21·43-s + 1.44·48-s + 10/7·49-s + 1.12·51-s − 2.49·52-s − 1.64·53-s + 2.56·61-s + 3/8·64-s + 1.45·68-s − 3.85·69-s − 0.230·75-s − 1.80·79-s + 5/9·81-s + 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.468836196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.468836196\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−12.72732018640175976671471463221, −12.10092389946497967094518845263, −11.73357396073929429459945207173, −11.69474380175466232146780880744, −10.52223744335013142535052849493, −10.26500708952233229644507372900, −9.816589303913704214365748237135, −9.565906579272753576393468975562, −8.480145042370734818966805905501, −8.187579500563309076026877242675, −7.59174351985760416099526883224, −7.34216665971660701418224122869, −6.67437241435915524858130345297, −6.08686581816896391275181852712, −5.45547671157826123522163666875, −4.52668775556950447886645155713, −3.78394649756968781007067772104, −3.03245265967709586286349457604, −2.30089519745610317539094762195, −1.86683055293904426504490175266,
1.86683055293904426504490175266, 2.30089519745610317539094762195, 3.03245265967709586286349457604, 3.78394649756968781007067772104, 4.52668775556950447886645155713, 5.45547671157826123522163666875, 6.08686581816896391275181852712, 6.67437241435915524858130345297, 7.34216665971660701418224122869, 7.59174351985760416099526883224, 8.187579500563309076026877242675, 8.480145042370734818966805905501, 9.565906579272753576393468975562, 9.816589303913704214365748237135, 10.26500708952233229644507372900, 10.52223744335013142535052849493, 11.69474380175466232146780880744, 11.73357396073929429459945207173, 12.10092389946497967094518845263, 12.72732018640175976671471463221
