| L(s) = 1 | + 2·2-s + 4-s − 4·7-s − 2·8-s + 2·9-s − 8·14-s − 3·16-s + 4·18-s − 2·25-s − 4·28-s − 8·29-s + 2·32-s + 2·36-s + 4·37-s − 43-s + 9·49-s − 4·50-s + 24·53-s + 8·56-s − 16·58-s − 8·63-s + 9·64-s − 4·67-s − 4·72-s + 8·74-s + 12·79-s − 5·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 1.51·7-s − 0.707·8-s + 2/3·9-s − 2.13·14-s − 3/4·16-s + 0.942·18-s − 2/5·25-s − 0.755·28-s − 1.48·29-s + 0.353·32-s + 1/3·36-s + 0.657·37-s − 0.152·43-s + 9/7·49-s − 0.565·50-s + 3.29·53-s + 1.06·56-s − 2.10·58-s − 1.00·63-s + 9/8·64-s − 0.488·67-s − 0.471·72-s + 0.929·74-s + 1.35·79-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4214 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4214 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.253347774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253347774\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−12.69399872525519535395988403832, −12.05661479780604306913851092700, −11.65319035564256849272441040770, −10.65137674321347955886098328352, −10.09842956724269002358647479802, −9.447636832833358231341172547892, −9.059315227315304168950227025977, −8.022130933086717285809494733001, −7.15004298714692546941344171843, −6.56601674599390260914140887948, −5.81003040597629193035206778203, −5.27313854727823898814799218586, −4.09067499425749664316975947210, −3.76873601533378217438002674930, −2.68529452850050109715166194854,
2.68529452850050109715166194854, 3.76873601533378217438002674930, 4.09067499425749664316975947210, 5.27313854727823898814799218586, 5.81003040597629193035206778203, 6.56601674599390260914140887948, 7.15004298714692546941344171843, 8.022130933086717285809494733001, 9.059315227315304168950227025977, 9.447636832833358231341172547892, 10.09842956724269002358647479802, 10.65137674321347955886098328352, 11.65319035564256849272441040770, 12.05661479780604306913851092700, 12.69399872525519535395988403832
