| L(s) = 1 | + 2·5-s + 16-s + 25-s + 4·49-s + 2·80-s + 81-s − 4·121-s − 2·125-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 + ⋯ |
| L(s) = 1 | + 2·5-s + 16-s + 25-s + 4·49-s + 2·80-s + 81-s − 4·121-s − 2·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.116807416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.116807416\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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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
−6.60050261349348357503519818599, −6.47797483720530103212368802050, −6.35407169275257244044472555794, −6.27074304660760730083286023088, −5.88723929792624444368846182843, −5.67072015497040851801477986229, −5.64480108074740047496686458861, −5.29154974167629111075630208200, −5.21121631345087000205805225512, −5.01897227222610891168895507691, −4.84828246078730541483681781557, −4.22469595122607518637532847040, −4.17086223661143964395487298353, −3.90595830391495515150895757561, −3.85777897993267101066853400577, −3.45198489762486305294732973571, −3.16841933016485343133666583323, −2.70744949697791368295009133013, −2.61579897272073265621530696101, −2.45602797975260126361993665697, −2.11150429836783181531835637608, −1.86904023238677044896944476337, −1.34575692393145751466489546728, −1.31679355163291657121485635824, −0.822938222305756065845047345046,
0.822938222305756065845047345046, 1.31679355163291657121485635824, 1.34575692393145751466489546728, 1.86904023238677044896944476337, 2.11150429836783181531835637608, 2.45602797975260126361993665697, 2.61579897272073265621530696101, 2.70744949697791368295009133013, 3.16841933016485343133666583323, 3.45198489762486305294732973571, 3.85777897993267101066853400577, 3.90595830391495515150895757561, 4.17086223661143964395487298353, 4.22469595122607518637532847040, 4.84828246078730541483681781557, 5.01897227222610891168895507691, 5.21121631345087000205805225512, 5.29154974167629111075630208200, 5.64480108074740047496686458861, 5.67072015497040851801477986229, 5.88723929792624444368846182843, 6.27074304660760730083286023088, 6.35407169275257244044472555794, 6.47797483720530103212368802050, 6.60050261349348357503519818599
