| L(s) = 1 | + 10·9-s − 12·17-s − 36·41-s − 4·49-s − 28·73-s + 57·81-s − 12·89-s − 56·97-s + 60·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 120·153-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 10/3·9-s − 2.91·17-s − 5.62·41-s − 4/7·49-s − 3.27·73-s + 19/3·81-s − 1.27·89-s − 5.68·97-s + 5.64·113-s + 2.36·121-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 − 9.70·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2705440488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2705440488\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 59 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
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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
−6.91088661987878728805418645202, −6.47112624282762758891446549118, −6.46067345777764791777940945302, −6.13087382262574477057900967854, −5.98089814944552536976952890433, −5.65160285619789485509194899951, −5.15638641956372212996978496981, −5.11884345635840674063961106964, −4.88584051642711154660293347326, −4.71955274260915204150013817709, −4.47945833697788903812045583287, −4.30987531222875992573776096024, −4.05116931983957084273570151249, −3.81980025532918971019684572588, −3.79675474619812289015271751798, −3.18804795881932844336566795589, −3.08521673887515311622077937819, −2.81916895462776555185454130216, −2.36693742157622901068982187610, −1.89303465965727662156161528727, −1.80355679736556479217145312881, −1.62026431824694552832215749993, −1.47973081456259457162912681181, −0.876262531421629958077602881430, −0.093678746395081186256233238952,
0.093678746395081186256233238952, 0.876262531421629958077602881430, 1.47973081456259457162912681181, 1.62026431824694552832215749993, 1.80355679736556479217145312881, 1.89303465965727662156161528727, 2.36693742157622901068982187610, 2.81916895462776555185454130216, 3.08521673887515311622077937819, 3.18804795881932844336566795589, 3.79675474619812289015271751798, 3.81980025532918971019684572588, 4.05116931983957084273570151249, 4.30987531222875992573776096024, 4.47945833697788903812045583287, 4.71955274260915204150013817709, 4.88584051642711154660293347326, 5.11884345635840674063961106964, 5.15638641956372212996978496981, 5.65160285619789485509194899951, 5.98089814944552536976952890433, 6.13087382262574477057900967854, 6.46067345777764791777940945302, 6.47112624282762758891446549118, 6.91088661987878728805418645202
