L(s) = 1 | + 4·13-s − 4·25-s − 4·49-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-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 + 241-s + ⋯ |
L(s) = 1 | + 4·13-s − 4·25-s − 4·49-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-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 + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.711175208\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.711175208\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
good | 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.33248571801848505585409516999, −5.89186004332812918733570909831, −5.88497423548763766168385900960, −5.73318104521047876448059632311, −5.46669440232052679027970366996, −5.29308371915132651551587751796, −5.21063137483275099168203687998, −4.72516415845926324803587653670, −4.44749028957424214909209794350, −4.25836106705058024385163749480, −4.16999996962144255646405830890, −4.12527497364545517259097100481, −3.67461667684720974056528045528, −3.43442285769496290198143521626, −3.37128124558202806589129757407, −3.26218948561992834014558808141, −3.12747041926156170629589083624, −2.49891886504529678394497968118, −2.40016820941047323624451859989, −1.79049215104578097119088190204, −1.76662863656922396801212754650, −1.71997506190771126393735826332, −1.40337545614928949919051619408, −0.963887853513604646285638321325, −0.49703993500675149397681882914,
0.49703993500675149397681882914, 0.963887853513604646285638321325, 1.40337545614928949919051619408, 1.71997506190771126393735826332, 1.76662863656922396801212754650, 1.79049215104578097119088190204, 2.40016820941047323624451859989, 2.49891886504529678394497968118, 3.12747041926156170629589083624, 3.26218948561992834014558808141, 3.37128124558202806589129757407, 3.43442285769496290198143521626, 3.67461667684720974056528045528, 4.12527497364545517259097100481, 4.16999996962144255646405830890, 4.25836106705058024385163749480, 4.44749028957424214909209794350, 4.72516415845926324803587653670, 5.21063137483275099168203687998, 5.29308371915132651551587751796, 5.46669440232052679027970366996, 5.73318104521047876448059632311, 5.88497423548763766168385900960, 5.89186004332812918733570909831, 6.33248571801848505585409516999