L(s) = 1 | + 4·3-s + 10·9-s − 16-s − 4·17-s + 20·27-s + 4·43-s − 4·48-s − 16·51-s + 35·81-s − 4·107-s + 4·113-s − 4·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s − 40·153-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 4·3-s + 10·9-s − 16-s − 4·17-s + 20·27-s + 4·43-s − 4·48-s − 16·51-s + 35·81-s − 4·107-s + 4·113-s − 4·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s − 40·153-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(5.539317858\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.539317858\) |
\(L(1)\) |
|
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
−7.01842154701136052638403789465, −6.83189348450268094654289111670, −6.82638918579746069196331755961, −6.49806366025205173377229351045, −6.06013572796187894318966208693, −5.97859421161292538745308059314, −5.95858762495327392133573555824, −4.98146631992474167106096851895, −4.94805255767929023489199281854, −4.91908464627980076951753285388, −4.54211115151797703555642602882, −4.33158477315213165976867180788, −4.03957937575070272124451128453, −4.03450169591669412052882930906, −3.87653230167456957326016644601, −3.57578624758234445644758624542, −3.12210891156062601440413911045, −2.96753746262196121044430277518, −2.43682870814301258940203836563, −2.40121690528520517899430977326, −2.39858619384828046134713780174, −2.26622689572498962660598281407, −1.81470004441514131644055188475, −1.25763938007383682012569400836, −1.19233075462391775007203047007,
1.19233075462391775007203047007, 1.25763938007383682012569400836, 1.81470004441514131644055188475, 2.26622689572498962660598281407, 2.39858619384828046134713780174, 2.40121690528520517899430977326, 2.43682870814301258940203836563, 2.96753746262196121044430277518, 3.12210891156062601440413911045, 3.57578624758234445644758624542, 3.87653230167456957326016644601, 4.03450169591669412052882930906, 4.03957937575070272124451128453, 4.33158477315213165976867180788, 4.54211115151797703555642602882, 4.91908464627980076951753285388, 4.94805255767929023489199281854, 4.98146631992474167106096851895, 5.95858762495327392133573555824, 5.97859421161292538745308059314, 6.06013572796187894318966208693, 6.49806366025205173377229351045, 6.82638918579746069196331755961, 6.83189348450268094654289111670, 7.01842154701136052638403789465