| L(s) = 1 | − 4·3-s − 2·4-s + 10·9-s + 8·12-s + 6·13-s + 3·16-s − 2·17-s − 2·23-s + 7·25-s − 20·27-s − 2·29-s − 20·36-s − 24·39-s − 34·43-s − 12·48-s − 2·49-s + 8·51-s − 12·52-s + 36·53-s + 2·61-s − 4·64-s + 4·68-s + 8·69-s − 28·75-s + 64·79-s + 35·81-s + 8·87-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 4-s + 10/3·9-s + 2.30·12-s + 1.66·13-s + 3/4·16-s − 0.485·17-s − 0.417·23-s + 7/5·25-s − 3.84·27-s − 0.371·29-s − 3.33·36-s − 3.84·39-s − 5.18·43-s − 1.73·48-s − 2/7·49-s + 1.12·51-s − 1.66·52-s + 4.94·53-s + 0.256·61-s − 1/2·64-s + 0.485·68-s + 0.963·69-s − 3.23·75-s + 7.20·79-s + 35/9·81-s + 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7391335349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7391335349\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - p T^{2} + 64 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 67 T^{2} + 1840 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 20 T^{2} + 934 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 31 T^{2} - 120 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 17 T + 154 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 3814 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 3766 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - T + 118 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 216 T^{2} + 20030 T^{4} - 216 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 4974 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 103 T^{2} + 11776 T^{4} - 103 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.010271066408051942881125907744, −7.32072216481566907410537520651, −7.17424229232181877757388618748, −6.97364234987324441591874111064, −6.70208531573655843371257144585, −6.61875524705064131753866320282, −6.27230941465201258216522956180, −6.17339577438278904287078365950, −5.69016212473853088460661458379, −5.60423596972005402576376875847, −5.36164933279997614873877654856, −5.10285295172301647332546109227, −4.71893763562569546200998891954, −4.71037456134547220526955053946, −4.58406684446153368875205253239, −3.82052227468415415111274129880, −3.64149204114418432137774574141, −3.63539967962175318920647713713, −3.46453253255970694580335035908, −2.62615273599389016373788112315, −2.19407707512459937400723072633, −1.80185961144397048832004952143, −1.31355843075987360880619276384, −0.834707819000932310465833839667, −0.47871636239499742434539236738,
0.47871636239499742434539236738, 0.834707819000932310465833839667, 1.31355843075987360880619276384, 1.80185961144397048832004952143, 2.19407707512459937400723072633, 2.62615273599389016373788112315, 3.46453253255970694580335035908, 3.63539967962175318920647713713, 3.64149204114418432137774574141, 3.82052227468415415111274129880, 4.58406684446153368875205253239, 4.71037456134547220526955053946, 4.71893763562569546200998891954, 5.10285295172301647332546109227, 5.36164933279997614873877654856, 5.60423596972005402576376875847, 5.69016212473853088460661458379, 6.17339577438278904287078365950, 6.27230941465201258216522956180, 6.61875524705064131753866320282, 6.70208531573655843371257144585, 6.97364234987324441591874111064, 7.17424229232181877757388618748, 7.32072216481566907410537520651, 8.010271066408051942881125907744
