| L(s) = 1 | − 2·2-s + 3·4-s − 5-s − 4·8-s + 2·9-s + 2·10-s + 13-s + 5·16-s − 17-s − 4·18-s − 3·20-s − 2·26-s + 29-s − 6·32-s + 2·34-s + 6·36-s + 37-s + 4·40-s + 41-s − 2·45-s + 2·49-s + 3·52-s + 53-s − 2·58-s − 61-s + 7·64-s − 65-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 5-s − 4·8-s + 2·9-s + 2·10-s + 13-s + 5·16-s − 17-s − 4·18-s − 3·20-s − 2·26-s + 29-s − 6·32-s + 2·34-s + 6·36-s + 37-s + 4·40-s + 41-s − 2·45-s + 2·49-s + 3·52-s + 53-s − 2·58-s − 61-s + 7·64-s − 65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4857424703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4857424703\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 17 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 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
−9.689242347296838994365521450259, −9.686355119728543959019669599620, −8.895717640035701762689097646290, −8.878353664731854470088932679646, −8.383454299978431655208036264200, −7.928169030090484556286769673947, −7.50724612509392176619659619870, −7.31510960255196794872843257777, −6.97024317239927086920734796609, −6.49680650443331888950116910231, −6.04675445483723202641328081025, −5.76541842432722038985123047858, −4.67114994899042061909186054176, −4.44979234967943188284752610360, −3.68880532658430484676578010246, −3.55170664133589650237045826662, −2.50894823879399534344941514697, −2.23037934588408357862289591024, −1.31287046705749264617439250942, −0.927496171838661277290974154215,
0.927496171838661277290974154215, 1.31287046705749264617439250942, 2.23037934588408357862289591024, 2.50894823879399534344941514697, 3.55170664133589650237045826662, 3.68880532658430484676578010246, 4.44979234967943188284752610360, 4.67114994899042061909186054176, 5.76541842432722038985123047858, 6.04675445483723202641328081025, 6.49680650443331888950116910231, 6.97024317239927086920734796609, 7.31510960255196794872843257777, 7.50724612509392176619659619870, 7.928169030090484556286769673947, 8.383454299978431655208036264200, 8.878353664731854470088932679646, 8.895717640035701762689097646290, 9.686355119728543959019669599620, 9.689242347296838994365521450259
