| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s − 14-s + 15-s − 16-s − 21-s + 23-s − 24-s + 27-s + 29-s − 30-s − 35-s − 40-s + 41-s + 42-s − 2·43-s − 46-s + 47-s + 48-s + 49-s − 54-s + 56-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s − 14-s + 15-s − 16-s − 21-s + 23-s − 24-s + 27-s + 29-s − 30-s − 35-s − 40-s + 41-s + 42-s − 2·43-s − 46-s + 47-s + 48-s + 49-s − 54-s + 56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1855957153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1855957153\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−13.04264661200903517342027650101, −12.42240255709205703010474074280, −11.81595518153823601415363561867, −11.73600259250111435577981040744, −10.91868286944090228996622822922, −10.90274568229085661893260839840, −10.36180860361938804868043823832, −9.612274076556058093848870083957, −9.144102951616797963774964463606, −8.310140351924579553413081198973, −8.286011076617848398893161331147, −7.72987037568855243508650243723, −6.85589382162551048678061698581, −6.73967792274990793223112557608, −5.44771769571650859231311693028, −5.24204552749825469170490668341, −4.43443877732487047943431891381, −3.96139403068010698317325041094, −2.69175475848050787785423695874, −1.22605061796298853307315365905,
1.22605061796298853307315365905, 2.69175475848050787785423695874, 3.96139403068010698317325041094, 4.43443877732487047943431891381, 5.24204552749825469170490668341, 5.44771769571650859231311693028, 6.73967792274990793223112557608, 6.85589382162551048678061698581, 7.72987037568855243508650243723, 8.286011076617848398893161331147, 8.310140351924579553413081198973, 9.144102951616797963774964463606, 9.612274076556058093848870083957, 10.36180860361938804868043823832, 10.90274568229085661893260839840, 10.91868286944090228996622822922, 11.73600259250111435577981040744, 11.81595518153823601415363561867, 12.42240255709205703010474074280, 13.04264661200903517342027650101
