| L(s) = 1 | + 40·25-s − 112·49-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 241-s + 251-s + 13·256-s + ⋯ |
| L(s) = 1 | + 8·25-s − 16·49-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + 0.812·256-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01737270053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01737270053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 13 T^{8} + p^{8} T^{16} \) |
| 5 | \( ( 1 - p T^{2} )^{8} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| good | 3 | \( ( 1 + 142 T^{8} + p^{8} T^{16} )^{2} \) |
| 7 | \( ( 1 + p T^{2} )^{16} \) |
| 11 | \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 + 52622 T^{8} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 - p T^{2} )^{16} \) |
| 23 | \( ( 1 + p T^{2} )^{16} \) |
| 29 | \( ( 1 - p T^{2} )^{16} \) |
| 31 | \( ( 1 + p T^{2} )^{16} \) |
| 37 | \( ( 1 - 3237298 T^{8} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 - p T^{2} )^{16} \) |
| 43 | \( ( 1 + p T^{2} )^{16} \) |
| 47 | \( ( 1 + p T^{2} )^{16} \) |
| 53 | \( ( 1 + 15154382 T^{8} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 + p T^{2} )^{16} \) |
| 61 | \( ( 1 - 7138 T^{4} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 + 3364622 T^{8} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 + p T^{2} )^{16} \) |
| 73 | \( ( 1 - p T^{2} )^{16} \) |
| 79 | \( ( 1 + p T^{2} )^{16} \) |
| 83 | \( ( 1 + p T^{2} )^{16} \) |
| 89 | \( ( 1 - p T^{2} )^{16} \) |
| 97 | \( ( 1 - 60577618 T^{8} + p^{8} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.09728109222089214583706140783, −3.08240191714102145056890987587, −3.07774613099976410096988920514, −3.05662483446825610801703526235, −2.82022953611963182368969079772, −2.81616788168418079695592099185, −2.79304424037988609126098550892, −2.77645644283675501732146163307, −2.41864101910281138285851983772, −2.35078115572950743095300862436, −2.16908196039831472953955375367, −2.02316253771729066816255870837, −2.01155673305768948380695139989, −1.83813499744971394423245195228, −1.82147832172138193677839087329, −1.68776458509567614087402177296, −1.49868430496705304224073561224, −1.38777519347960803787656745516, −1.36269486951045195331853012536, −1.28001129111446408026241840479, −1.11020126310432041625874057187, −0.923751885223943354440011888904, −0.72055009389670173550359412257, −0.44213399165821938233198961366, −0.01523713701224084538483115035,
0.01523713701224084538483115035, 0.44213399165821938233198961366, 0.72055009389670173550359412257, 0.923751885223943354440011888904, 1.11020126310432041625874057187, 1.28001129111446408026241840479, 1.36269486951045195331853012536, 1.38777519347960803787656745516, 1.49868430496705304224073561224, 1.68776458509567614087402177296, 1.82147832172138193677839087329, 1.83813499744971394423245195228, 2.01155673305768948380695139989, 2.02316253771729066816255870837, 2.16908196039831472953955375367, 2.35078115572950743095300862436, 2.41864101910281138285851983772, 2.77645644283675501732146163307, 2.79304424037988609126098550892, 2.81616788168418079695592099185, 2.82022953611963182368969079772, 3.05662483446825610801703526235, 3.07774613099976410096988920514, 3.08240191714102145056890987587, 3.09728109222089214583706140783
Plot not available for L-functions of degree greater than 10.
