| L(s) = 1 | − 4·3-s − 2·4-s + 10·9-s + 8·12-s + 3·16-s + 12·23-s + 14·25-s − 20·27-s − 12·29-s − 20·36-s − 4·43-s − 12·48-s + 4·49-s + 12·53-s + 40·61-s − 4·64-s − 48·69-s − 56·75-s − 8·79-s + 35·81-s + 48·87-s − 24·92-s − 28·100-s − 36·101-s − 4·103-s + 12·107-s + 40·108-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 4-s + 10/3·9-s + 2.30·12-s + 3/4·16-s + 2.50·23-s + 14/5·25-s − 3.84·27-s − 2.22·29-s − 3.33·36-s − 0.609·43-s − 1.73·48-s + 4/7·49-s + 1.64·53-s + 5.12·61-s − 1/2·64-s − 5.77·69-s − 6.46·75-s − 0.900·79-s + 35/9·81-s + 5.14·87-s − 2.50·92-s − 2.79·100-s − 3.58·101-s − 0.394·103-s + 1.16·107-s + 3.84·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.212906865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212906865\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 195 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 20346 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 17802 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.03286715841804917676065692512, −6.96992845859822523781255341958, −6.69010141680569072618755070945, −6.63781016951225988056677613452, −6.10759920385445953389950688821, −5.85971115654477828802436150836, −5.62935354024787477675599788695, −5.46976260484627359185834123977, −5.44713705908240104729018003588, −5.04083237111108977688038307101, −4.94674985340704599099193371863, −4.61998220779201820251734985220, −4.61501979001277423899587267339, −4.06580538218101753146201116042, −4.01095630876779920814389310177, −3.63435942592410133375442644641, −3.35315755462808610305718408534, −3.15759422700919817152070249273, −2.63377374998890150958925067877, −2.35480109575313929094526462624, −1.97184866047405320612611993334, −1.33245491059348293589000806134, −1.15608992486730674946899329763, −0.62353040943191588381020854108, −0.56983721854459860524711975786,
0.56983721854459860524711975786, 0.62353040943191588381020854108, 1.15608992486730674946899329763, 1.33245491059348293589000806134, 1.97184866047405320612611993334, 2.35480109575313929094526462624, 2.63377374998890150958925067877, 3.15759422700919817152070249273, 3.35315755462808610305718408534, 3.63435942592410133375442644641, 4.01095630876779920814389310177, 4.06580538218101753146201116042, 4.61501979001277423899587267339, 4.61998220779201820251734985220, 4.94674985340704599099193371863, 5.04083237111108977688038307101, 5.44713705908240104729018003588, 5.46976260484627359185834123977, 5.62935354024787477675599788695, 5.85971115654477828802436150836, 6.10759920385445953389950688821, 6.63781016951225988056677613452, 6.69010141680569072618755070945, 6.96992845859822523781255341958, 7.03286715841804917676065692512
