| L(s) = 1 | + 2·3-s + 14·7-s − 5·9-s − 50·13-s + 14·19-s + 28·21-s − 28·27-s + 14·31-s + 4·37-s − 100·39-s − 82·43-s + 49·49-s + 28·57-s − 2·61-s − 70·63-s − 34·67-s − 140·73-s + 116·79-s − 11·81-s − 700·91-s + 28·93-s − 98·97-s + 308·103-s − 50·109-s + 8·111-s + 250·117-s + 170·121-s + ⋯ |
| L(s) = 1 | + 2/3·3-s + 2·7-s − 5/9·9-s − 3.84·13-s + 0.736·19-s + 4/3·21-s − 1.03·27-s + 0.451·31-s + 4/37·37-s − 2.56·39-s − 1.90·43-s + 49-s + 0.491·57-s − 0.0327·61-s − 1.11·63-s − 0.507·67-s − 1.91·73-s + 1.46·79-s − 0.135·81-s − 7.69·91-s + 0.301·93-s − 1.01·97-s + 2.99·103-s − 0.458·109-s + 0.0720·111-s + 2.13·117-s + 1.40·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.745635425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.745635425\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 170 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 70 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 410 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 118 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 41 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2090 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8282 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 334 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2590 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 49 T + p^{2} T^{2} )^{2} \) |
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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
−9.654369544641867486065104388231, −9.555105942634100894442716820632, −8.757383274047098096649485132603, −8.594702413198485034848673343216, −7.902510608314406486550306029786, −7.86690159070136632162818382135, −7.42847093444129848156348725171, −7.12138683347527511915717219109, −6.62009533280546283297034795974, −5.74585019278562682775698971892, −5.42038965846699785731010433713, −4.86385638206789412012123959107, −4.71447875438696003659017012855, −4.46878811052058189696630149089, −3.38616780169582751190589176500, −3.02377698141610002159515607062, −2.29869132523221975749581522045, −2.14737934543105428712629867665, −1.46759905849771493921514604054, −0.35354229502205263785839925885,
0.35354229502205263785839925885, 1.46759905849771493921514604054, 2.14737934543105428712629867665, 2.29869132523221975749581522045, 3.02377698141610002159515607062, 3.38616780169582751190589176500, 4.46878811052058189696630149089, 4.71447875438696003659017012855, 4.86385638206789412012123959107, 5.42038965846699785731010433713, 5.74585019278562682775698971892, 6.62009533280546283297034795974, 7.12138683347527511915717219109, 7.42847093444129848156348725171, 7.86690159070136632162818382135, 7.902510608314406486550306029786, 8.594702413198485034848673343216, 8.757383274047098096649485132603, 9.555105942634100894442716820632, 9.654369544641867486065104388231
