| 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 − 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 − 16·79-s + 5·81-s − 8·87-s + 48·92-s − 4·101-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 − 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 − 1.80·79-s + 5/9·81-s − 0.857·87-s + 5.00·92-s − 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.743151328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.743151328\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 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
−10.37175477252866260186786002138, −10.13336608059775977039909759168, −9.250026521604170608471763914047, −9.073911143958808428783703376788, −8.449525282304171456462960788926, −8.276703869195701940548040106465, −7.18786716813904239023266219348, −7.18130209047634196329054430817, −6.88422917842219885207653566343, −6.48132365688038266589261786180, −5.85380573295926455919206720937, −5.70165687914197328129668958458, −5.12868739012297732474610252066, −4.55238946818830063525665263081, −3.98059232606463378037360849824, −3.42702303067957285369968240427, −2.66277753723284712249708639345, −2.34710161549458293039409910544, −1.17492805803391975860218363320, −1.04007413228782745627927839378,
1.04007413228782745627927839378, 1.17492805803391975860218363320, 2.34710161549458293039409910544, 2.66277753723284712249708639345, 3.42702303067957285369968240427, 3.98059232606463378037360849824, 4.55238946818830063525665263081, 5.12868739012297732474610252066, 5.70165687914197328129668958458, 5.85380573295926455919206720937, 6.48132365688038266589261786180, 6.88422917842219885207653566343, 7.18130209047634196329054430817, 7.18786716813904239023266219348, 8.276703869195701940548040106465, 8.449525282304171456462960788926, 9.073911143958808428783703376788, 9.250026521604170608471763914047, 10.13336608059775977039909759168, 10.37175477252866260186786002138
