| L(s) = 1 | + 2·3-s + 3·9-s + 2·11-s − 2·19-s − 25-s + 6·27-s + 4·33-s + 2·41-s + 2·43-s − 2·49-s − 4·57-s − 2·67-s + 2·73-s − 2·75-s + 9·81-s + 2·97-s + 6·99-s + 2·113-s + 3·121-s + 4·123-s + 127-s + 4·129-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + ⋯ |
| L(s) = 1 | + 2·3-s + 3·9-s + 2·11-s − 2·19-s − 25-s + 6·27-s + 4·33-s + 2·41-s + 2·43-s − 2·49-s − 4·57-s − 2·67-s + 2·73-s − 2·75-s + 9·81-s + 2·97-s + 6·99-s + 2·113-s + 3·121-s + 4·123-s + 127-s + 4·129-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.037979487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.037979487\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.53965605888150107162065506616, −6.12695205021190603148704099688, −6.10427414464888157679500699422, −6.01255922708122516831342151166, −5.86802120836177946739421512763, −5.09408692557147401448354601100, −5.06204406608334819910012405669, −4.95221943254240720503036446232, −4.62416306669467965755958430644, −4.41881785069666479594765214681, −4.26507641774063281630102816012, −4.10407250123421276693561503817, −3.75856571159128602131139794255, −3.67846858697648895926639916634, −3.67165800867061436642356626256, −3.07965008599347670591814628963, −2.95507095246583913182854027884, −2.81509808379246081577271584049, −2.35131818608718949373813984806, −2.26162059233679385888283944106, −2.02488126348234747902788692802, −1.78435383028287559556780329888, −1.32146734246307972710866374836, −1.11713360638909382170224689425, −0.878020261799088288070335889632,
0.878020261799088288070335889632, 1.11713360638909382170224689425, 1.32146734246307972710866374836, 1.78435383028287559556780329888, 2.02488126348234747902788692802, 2.26162059233679385888283944106, 2.35131818608718949373813984806, 2.81509808379246081577271584049, 2.95507095246583913182854027884, 3.07965008599347670591814628963, 3.67165800867061436642356626256, 3.67846858697648895926639916634, 3.75856571159128602131139794255, 4.10407250123421276693561503817, 4.26507641774063281630102816012, 4.41881785069666479594765214681, 4.62416306669467965755958430644, 4.95221943254240720503036446232, 5.06204406608334819910012405669, 5.09408692557147401448354601100, 5.86802120836177946739421512763, 6.01255922708122516831342151166, 6.10427414464888157679500699422, 6.12695205021190603148704099688, 6.53965605888150107162065506616
