| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3977985808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3977985808\) |
| \(L(1)\) |
|
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
−13.13201990331895685581857017512, −12.90053953947723771936297872153, −12.47657713772994581670433605768, −12.16504101055456686389956090746, −11.48777106355019723346687957075, −10.84845903149344503222006345713, −10.64344170416799654257529046120, −9.882984495462040023396895619130, −9.557610199895939780519752160263, −9.277947762854317866051502400256, −8.022162073069212996887088265641, −7.20928712881975974576438377952, −6.72697108850936096934595131411, −6.41241361601194481862521675836, −5.52480129567178177678679215285, −5.40933860747215012818351415228, −4.72930641781261513377891837575, −4.21212687084065104754043933071, −3.21821698862269073260840566227, −1.81866389132722042253009866904,
1.81866389132722042253009866904, 3.21821698862269073260840566227, 4.21212687084065104754043933071, 4.72930641781261513377891837575, 5.40933860747215012818351415228, 5.52480129567178177678679215285, 6.41241361601194481862521675836, 6.72697108850936096934595131411, 7.20928712881975974576438377952, 8.022162073069212996887088265641, 9.277947762854317866051502400256, 9.557610199895939780519752160263, 9.882984495462040023396895619130, 10.64344170416799654257529046120, 10.84845903149344503222006345713, 11.48777106355019723346687957075, 12.16504101055456686389956090746, 12.47657713772994581670433605768, 12.90053953947723771936297872153, 13.13201990331895685581857017512
