L(s) = 1 | + 2·2-s − 4-s + 2·7-s − 8·8-s − 2·9-s + 6·11-s + 4·14-s − 7·16-s − 4·18-s + 12·22-s + 6·23-s + 6·25-s − 2·28-s − 29-s + 14·32-s + 2·36-s + 9·37-s + 10·43-s − 6·44-s + 12·46-s − 3·49-s + 12·50-s − 16·53-s − 16·56-s − 2·58-s − 4·63-s + 35·64-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1/2·4-s + 0.755·7-s − 2.82·8-s − 2/3·9-s + 1.80·11-s + 1.06·14-s − 7/4·16-s − 0.942·18-s + 2.55·22-s + 1.25·23-s + 6/5·25-s − 0.377·28-s − 0.185·29-s + 2.47·32-s + 1/3·36-s + 1.47·37-s + 1.52·43-s − 0.904·44-s + 1.76·46-s − 3/7·49-s + 1.69·50-s − 2.19·53-s − 2.13·56-s − 0.262·58-s − 0.503·63-s + 35/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52577 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52577 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.119862798\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.119862798\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 8 T + p T^{2} ) \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 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
−9.820664850348081234891614508731, −9.399626143694155154260181765749, −9.040562005474192363112282650538, −8.680491037086751828200919916777, −8.152880765765543166116243051998, −7.40289234891815113610892678881, −6.46938597061423655470317609084, −6.27799388256951323994916174952, −5.53612859810115270856442526392, −5.03142782484252507689203998941, −4.45965127873207295620285599693, −4.12661942372409735500575837750, −3.31077157086122666744663912424, −2.78329204730495419488092025283, −1.11808100361648947328725121339,
1.11808100361648947328725121339, 2.78329204730495419488092025283, 3.31077157086122666744663912424, 4.12661942372409735500575837750, 4.45965127873207295620285599693, 5.03142782484252507689203998941, 5.53612859810115270856442526392, 6.27799388256951323994916174952, 6.46938597061423655470317609084, 7.40289234891815113610892678881, 8.152880765765543166116243051998, 8.680491037086751828200919916777, 9.040562005474192363112282650538, 9.399626143694155154260181765749, 9.820664850348081234891614508731