| L(s) = 1 | − 2·4-s − 4·5-s + 2·9-s + 16-s − 2·19-s + 8·20-s + 10·25-s − 4·31-s − 4·36-s + 2·41-s − 8·45-s − 2·49-s + 2·59-s + 2·64-s − 2·71-s + 4·76-s − 4·80-s + 81-s + 8·95-s − 20·100-s − 2·101-s + 4·109-s − 2·121-s + 8·124-s − 20·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s − 4·5-s + 2·9-s + 16-s − 2·19-s + 8·20-s + 10·25-s − 4·31-s − 4·36-s + 2·41-s − 8·45-s − 2·49-s + 2·59-s + 2·64-s − 2·71-s + 4·76-s − 4·80-s + 81-s + 8·95-s − 20·100-s − 2·101-s + 4·109-s − 2·121-s + 8·124-s − 20·125-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02104373539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02104373539\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−6.90462016108604592222881923589, −6.65536719358089949956861551685, −6.52276719572584919219797994821, −5.96584377366271131378868977925, −5.95422409772995167873100461901, −5.40221358783587108585128407428, −5.24593808881362731225251243419, −5.19125041171068482694003410426, −4.67103950865467401603428121480, −4.64562400595853249375696807432, −4.36435858605497184682018966515, −4.30714991163580986441291342295, −4.27295470097515021265917141660, −4.00234198780960797610982792804, −3.69114515041586001311617957503, −3.57966518409483424019074569902, −3.50894621568163748058224379140, −3.11049624973495091304319654380, −2.83158227449255033420691839775, −2.20580174082125034035189673830, −2.16995092867022748048347163322, −1.64572066895247679578381263913, −1.14651985467179431511887067888, −0.863139815421126946163196074024, −0.11276467011578664861026201337,
0.11276467011578664861026201337, 0.863139815421126946163196074024, 1.14651985467179431511887067888, 1.64572066895247679578381263913, 2.16995092867022748048347163322, 2.20580174082125034035189673830, 2.83158227449255033420691839775, 3.11049624973495091304319654380, 3.50894621568163748058224379140, 3.57966518409483424019074569902, 3.69114515041586001311617957503, 4.00234198780960797610982792804, 4.27295470097515021265917141660, 4.30714991163580986441291342295, 4.36435858605497184682018966515, 4.64562400595853249375696807432, 4.67103950865467401603428121480, 5.19125041171068482694003410426, 5.24593808881362731225251243419, 5.40221358783587108585128407428, 5.95422409772995167873100461901, 5.96584377366271131378868977925, 6.52276719572584919219797994821, 6.65536719358089949956861551685, 6.90462016108604592222881923589
