| L(s) = 1 | − 2·2-s + 4-s + 2·5-s + 2·8-s − 4·10-s + 2·11-s − 4·16-s + 2·17-s + 2·20-s − 4·22-s + 25-s + 2·29-s − 2·31-s + 2·32-s − 4·34-s + 4·40-s + 2·44-s − 2·50-s + 4·55-s − 4·58-s + 2·61-s + 4·62-s + 3·64-s − 2·67-s + 2·68-s − 8·80-s − 2·83-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 2·5-s + 2·8-s − 4·10-s + 2·11-s − 4·16-s + 2·17-s + 2·20-s − 4·22-s + 25-s + 2·29-s − 2·31-s + 2·32-s − 4·34-s + 4·40-s + 2·44-s − 2·50-s + 4·55-s − 4·58-s + 2·61-s + 4·62-s + 3·64-s − 2·67-s + 2·68-s − 8·80-s − 2·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4626801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4626801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7209577906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7209577906\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 239 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2^2$ | \( 1 + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−9.403900991727383454971185836199, −9.208704971631530789653962510550, −8.735792119741634860765562326784, −8.613284827665564718178877719368, −8.134508540993616110825691166980, −7.58930044922002904632021011796, −7.26809148398465079991793981317, −6.96484129443832487962415621855, −6.28936765790551897519371521509, −6.16191576010009351777978057046, −5.42078580743045591351431096127, −5.38510191980781384452550227640, −4.67576576351142655721747271211, −4.15390296872720246988521497798, −3.75070727266357459269629251941, −3.12570361017273950383304364602, −2.25127978319250414000309853719, −1.78027265371229092511307052832, −1.28936431270954069317179779520, −1.07026718400937328860511981678,
1.07026718400937328860511981678, 1.28936431270954069317179779520, 1.78027265371229092511307052832, 2.25127978319250414000309853719, 3.12570361017273950383304364602, 3.75070727266357459269629251941, 4.15390296872720246988521497798, 4.67576576351142655721747271211, 5.38510191980781384452550227640, 5.42078580743045591351431096127, 6.16191576010009351777978057046, 6.28936765790551897519371521509, 6.96484129443832487962415621855, 7.26809148398465079991793981317, 7.58930044922002904632021011796, 8.134508540993616110825691166980, 8.613284827665564718178877719368, 8.735792119741634860765562326784, 9.208704971631530789653962510550, 9.403900991727383454971185836199
