| L(s) = 1 | − 4·19-s + 2·49-s − 4·61-s + 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 4·19-s + 2·49-s − 4·61-s + 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8150495650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8150495650\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−6.51238460270033905146623495305, −6.50694258561531982445029001036, −6.50347751916732354917210509194, −5.99004745699615919943991739275, −5.89068762861157760291461271730, −5.63076908321124070471609486504, −5.38412833681878083567992811931, −5.33888141925683638682176174584, −4.71388620877052884621676361421, −4.67362916347042190253038986671, −4.47748150253517346951882482002, −4.46596650223500158213644188886, −4.12912090296278936926376274417, −3.81147327691642548807846911105, −3.68493575279538663600998276755, −3.39341794277616475597240913041, −3.13525931027800871266299781335, −2.79236654733543617383755313581, −2.43612157846312220841865952203, −2.40825720303266386130029002400, −1.95934878714665840016153863370, −1.86200313568443260690063170890, −1.58062512804029687945387432507, −0.994737123430730438835313106031, −0.46525003722821184951591825544,
0.46525003722821184951591825544, 0.994737123430730438835313106031, 1.58062512804029687945387432507, 1.86200313568443260690063170890, 1.95934878714665840016153863370, 2.40825720303266386130029002400, 2.43612157846312220841865952203, 2.79236654733543617383755313581, 3.13525931027800871266299781335, 3.39341794277616475597240913041, 3.68493575279538663600998276755, 3.81147327691642548807846911105, 4.12912090296278936926376274417, 4.46596650223500158213644188886, 4.47748150253517346951882482002, 4.67362916347042190253038986671, 4.71388620877052884621676361421, 5.33888141925683638682176174584, 5.38412833681878083567992811931, 5.63076908321124070471609486504, 5.89068762861157760291461271730, 5.99004745699615919943991739275, 6.50347751916732354917210509194, 6.50694258561531982445029001036, 6.51238460270033905146623495305
