| L(s) = 1 | − 2-s − 4-s + 8-s + 9-s + 13-s + 3·16-s − 17-s − 18-s + 13·19-s − 25-s − 26-s − 3·32-s + 34-s − 36-s − 13·38-s + 7·43-s − 12·47-s − 4·49-s + 50-s − 52-s + 9·53-s − 6·59-s − 5·64-s + 67-s + 68-s + 72-s − 13·76-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.353·8-s + 1/3·9-s + 0.277·13-s + 3/4·16-s − 0.242·17-s − 0.235·18-s + 2.98·19-s − 1/5·25-s − 0.196·26-s − 0.530·32-s + 0.171·34-s − 1/6·36-s − 2.10·38-s + 1.06·43-s − 1.75·47-s − 4/7·49-s + 0.141·50-s − 0.138·52-s + 1.23·53-s − 0.781·59-s − 5/8·64-s + 0.122·67-s + 0.121·68-s + 0.117·72-s − 1.49·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9826 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9826 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6281177032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6281177032\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 17 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
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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
−11.45348208430271910086170342009, −11.00715532667550191457422723038, −10.12280483548531153976211382113, −9.760830905424297148904403222407, −9.415708920103122708962942455669, −8.762479710761537556577158675946, −8.165098267083513910121660174385, −7.53158090016727587396334338201, −7.12282134213881485718518693486, −6.09230632889220187781516200996, −5.43568718834111703193747271496, −4.78609474389382711636947815606, −3.76595342561720626277078825555, −3.00149030201737212911794106451, −1.25979826367286519817628369979,
1.25979826367286519817628369979, 3.00149030201737212911794106451, 3.76595342561720626277078825555, 4.78609474389382711636947815606, 5.43568718834111703193747271496, 6.09230632889220187781516200996, 7.12282134213881485718518693486, 7.53158090016727587396334338201, 8.165098267083513910121660174385, 8.762479710761537556577158675946, 9.415708920103122708962942455669, 9.760830905424297148904403222407, 10.12280483548531153976211382113, 11.00715532667550191457422723038, 11.45348208430271910086170342009
