L(s) = 1 | − 2·9-s − 4·11-s − 16-s + 4·43-s − 4·59-s + 2·81-s + 4·83-s − 4·97-s + 8·99-s + 4·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | − 2·9-s − 4·11-s − 16-s + 4·43-s − 4·59-s + 2·81-s + 4·83-s − 4·97-s + 8·99-s + 4·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1260520096\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1260520096\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05592689375429114078977474895, −9.718940012970763598304128826386, −9.483287034266436882843707780577, −9.069770138442056382190040664788, −8.779608712820201062561950996831, −8.776340600704764084919352485455, −8.175611521551417626168374837597, −8.107939694813905132159077763354, −7.58492444758015215348649087670, −7.58372770452251988614410918593, −7.52382658591326990188196679549, −6.94952358543983213480756466490, −6.30848864215535830980669951229, −6.03618322896740243925109865513, −5.96652199153184866586418734347, −5.50596068933523817240724901115, −5.33526846793379661168496317593, −4.80109863993334326261410933505, −4.76271806022856407544235163714, −4.33544363119264524956986759978, −3.48611060907769416643412290216, −3.21670837177715010608350948468, −2.66511094688412949363541073762, −2.45018497680460612401004236085, −2.29087843910759297804616113393,
2.29087843910759297804616113393, 2.45018497680460612401004236085, 2.66511094688412949363541073762, 3.21670837177715010608350948468, 3.48611060907769416643412290216, 4.33544363119264524956986759978, 4.76271806022856407544235163714, 4.80109863993334326261410933505, 5.33526846793379661168496317593, 5.50596068933523817240724901115, 5.96652199153184866586418734347, 6.03618322896740243925109865513, 6.30848864215535830980669951229, 6.94952358543983213480756466490, 7.52382658591326990188196679549, 7.58372770452251988614410918593, 7.58492444758015215348649087670, 8.107939694813905132159077763354, 8.175611521551417626168374837597, 8.776340600704764084919352485455, 8.779608712820201062561950996831, 9.069770138442056382190040664788, 9.483287034266436882843707780577, 9.718940012970763598304128826386, 10.05592689375429114078977474895