| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 3·9-s + 3·11-s + 12-s + 16-s + 6·17-s + 3·18-s + 3·19-s − 3·22-s − 24-s + 4·25-s − 4·27-s − 32-s + 3·33-s − 6·34-s − 3·36-s − 3·38-s + 6·41-s + 3·44-s + 48-s − 5·49-s − 4·50-s + 6·51-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 9-s + 0.904·11-s + 0.288·12-s + 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.688·19-s − 0.639·22-s − 0.204·24-s + 4/5·25-s − 0.769·27-s − 0.176·32-s + 0.522·33-s − 1.02·34-s − 1/2·36-s − 0.486·38-s + 0.937·41-s + 0.452·44-s + 0.144·48-s − 5/7·49-s − 0.565·50-s + 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15488 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15488 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9645283652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9645283652\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 11 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−11.22481092115097880532319151646, −10.31764890144086659691916673308, −10.05769427711872302310186538986, −9.218548905796806236866440332166, −8.996281679825403717753154852326, −8.475751166922685324458270645573, −7.69439992072876897576259330858, −7.48082649266952646711930598072, −6.55418608812910012715537520733, −5.93176085256290377425805362629, −5.36289380694213020826486163582, −4.31445266665482237413470665573, −3.25550419144057054080416878556, −2.86039698545766773311700838681, −1.39840382486424383710551066259,
1.39840382486424383710551066259, 2.86039698545766773311700838681, 3.25550419144057054080416878556, 4.31445266665482237413470665573, 5.36289380694213020826486163582, 5.93176085256290377425805362629, 6.55418608812910012715537520733, 7.48082649266952646711930598072, 7.69439992072876897576259330858, 8.475751166922685324458270645573, 8.996281679825403717753154852326, 9.218548905796806236866440332166, 10.05769427711872302310186538986, 10.31764890144086659691916673308, 11.22481092115097880532319151646
