| L(s) = 1 | + 16·31-s + 56·61-s − 9·81-s + 44·121-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 + ⋯ |
| L(s) = 1 | + 2.87·31-s + 7.17·61-s − 81-s + 4·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.087427694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.087427694\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 2062 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 3214 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 5906 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 8542 T^{4} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 - 18814 T^{4} + p^{4} T^{8} \) |
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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
−8.395828040338956258713311183158, −8.353456367036923359828498620284, −8.271115255958706346250263979573, −7.86339715024447441911602339562, −7.50798738926167795514168174967, −7.06683226675601598557856900859, −7.03338182723814703896016506530, −6.76857922228371503159190183850, −6.65378478733536299864913578006, −6.02158778135602199514818276050, −5.89180492702692051383416284098, −5.82290054571889066703307141897, −5.29340370255185237460372487794, −5.00657973306111747453348303330, −4.67163413279996298194903414777, −4.62939666683130432817846544659, −3.97975442981480372751068180553, −3.87033357540568871028019178587, −3.51011191800166728322480057009, −3.10809987766291724972668468928, −2.59330614551833375819427227728, −2.34826722357148605880232686547, −2.07065228059845667830178901591, −1.14306251508691973050411521430, −0.799160189443931801462285114043,
0.799160189443931801462285114043, 1.14306251508691973050411521430, 2.07065228059845667830178901591, 2.34826722357148605880232686547, 2.59330614551833375819427227728, 3.10809987766291724972668468928, 3.51011191800166728322480057009, 3.87033357540568871028019178587, 3.97975442981480372751068180553, 4.62939666683130432817846544659, 4.67163413279996298194903414777, 5.00657973306111747453348303330, 5.29340370255185237460372487794, 5.82290054571889066703307141897, 5.89180492702692051383416284098, 6.02158778135602199514818276050, 6.65378478733536299864913578006, 6.76857922228371503159190183850, 7.03338182723814703896016506530, 7.06683226675601598557856900859, 7.50798738926167795514168174967, 7.86339715024447441911602339562, 8.271115255958706346250263979573, 8.353456367036923359828498620284, 8.395828040338956258713311183158
