| L(s) = 1 | − 2-s + 2·3-s − 5-s − 2·6-s − 7-s + 3·9-s + 10-s − 13-s + 14-s − 2·15-s − 3·18-s − 2·21-s − 23-s + 26-s + 4·27-s + 2·30-s + 32-s + 35-s − 37-s − 2·39-s − 41-s + 2·42-s − 43-s − 3·45-s + 46-s + 2·53-s − 4·54-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s − 5-s − 2·6-s − 7-s + 3·9-s + 10-s − 13-s + 14-s − 2·15-s − 3·18-s − 2·21-s − 23-s + 26-s + 4·27-s + 2·30-s + 32-s + 35-s − 37-s − 2·39-s − 41-s + 2·42-s − 43-s − 3·45-s + 46-s + 2·53-s − 4·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3491217167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3491217167\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 53 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.29794649973420546914448632559, −13.12971081393997962784196767074, −12.31912059848638155757535555552, −12.08075452607637885321168741455, −11.50962232126074463883831350213, −10.37181457340143541192994701428, −9.976018845523336640746641565715, −9.929638345352002043042230505657, −9.140744998868018239527572375951, −8.827944736570786066227315986209, −8.319312957477228354448347977821, −7.985342886408146721932302617208, −7.16533614001499362983021333869, −7.12455000812335815902247706939, −6.18693376072904024118256695995, −4.92042080695935339238281285711, −4.19516455922442619181815882044, −3.56732910811294467104717462395, −2.99525620404322713807727056053, −2.06209860374741796339889994643,
2.06209860374741796339889994643, 2.99525620404322713807727056053, 3.56732910811294467104717462395, 4.19516455922442619181815882044, 4.92042080695935339238281285711, 6.18693376072904024118256695995, 7.12455000812335815902247706939, 7.16533614001499362983021333869, 7.985342886408146721932302617208, 8.319312957477228354448347977821, 8.827944736570786066227315986209, 9.140744998868018239527572375951, 9.929638345352002043042230505657, 9.976018845523336640746641565715, 10.37181457340143541192994701428, 11.50962232126074463883831350213, 12.08075452607637885321168741455, 12.31912059848638155757535555552, 13.12971081393997962784196767074, 13.29794649973420546914448632559
