| L(s) = 1 | − 5-s − 2·7-s − 9-s − 11-s − 2·13-s − 19-s − 23-s + 2·35-s + 37-s − 2·41-s + 45-s − 47-s + 3·49-s + 53-s + 55-s + 2·59-s + 2·63-s + 2·65-s + 2·77-s − 2·89-s + 4·91-s + 95-s + 99-s + 2·103-s + 115-s + 2·117-s + 121-s + ⋯ |
| L(s) = 1 | − 5-s − 2·7-s − 9-s − 11-s − 2·13-s − 19-s − 23-s + 2·35-s + 37-s − 2·41-s + 45-s − 47-s + 3·49-s + 53-s + 55-s + 2·59-s + 2·63-s + 2·65-s + 2·77-s − 2·89-s + 4·91-s + 95-s + 99-s + 2·103-s + 115-s + 2·117-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1200183986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1200183986\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−10.36763588970957441546389593178, −9.783567556443048594106761603668, −9.709131831028727635959237003598, −8.882849182052160919112437351062, −8.627004750211032363627211464254, −8.168754301810118842031985637516, −7.81078384865179824066837297089, −7.19385487108510001187299243654, −7.08758870790647450891163429356, −6.46173429197694358847415361560, −6.12500199597531378076261120364, −5.48047720458334428457373047470, −5.26396318446279451999506202372, −4.50105680923516349541086778094, −4.11616225416496281670907335725, −3.44798369180119781821639867289, −3.14245799167679114699953558451, −2.37962729814885197046488653328, −2.32529432360541342104995221585, −0.28562293778862235321863141056,
0.28562293778862235321863141056, 2.32529432360541342104995221585, 2.37962729814885197046488653328, 3.14245799167679114699953558451, 3.44798369180119781821639867289, 4.11616225416496281670907335725, 4.50105680923516349541086778094, 5.26396318446279451999506202372, 5.48047720458334428457373047470, 6.12500199597531378076261120364, 6.46173429197694358847415361560, 7.08758870790647450891163429356, 7.19385487108510001187299243654, 7.81078384865179824066837297089, 8.168754301810118842031985637516, 8.627004750211032363627211464254, 8.882849182052160919112437351062, 9.709131831028727635959237003598, 9.783567556443048594106761603668, 10.36763588970957441546389593178
