| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.908936844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.908936844\) |
| \(L(1)\) |
|
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
−10.02849723603225236203874858098, −9.788309088208788376460864511642, −9.165280137550611520540442374426, −8.628030587435983077100898783984, −7.915566344163055930494806306469, −7.70259492117001425523138851049, −7.19130116723388732080911721109, −7.17660430637350958581965305651, −6.60528638854424029309717881312, −6.38357226037683613251325512203, −5.48533769426723532315148175920, −5.35761959889671297849470314880, −4.62732332062340080010821326124, −4.48732510412639402399370797454, −3.97400568413910300898318330206, −3.80420260796449060522666033193, −3.12200734144396943976128537623, −2.52927368835307880466437208044, −1.87611444413021101515239433871, −1.46113443310373748222405680555,
1.46113443310373748222405680555, 1.87611444413021101515239433871, 2.52927368835307880466437208044, 3.12200734144396943976128537623, 3.80420260796449060522666033193, 3.97400568413910300898318330206, 4.48732510412639402399370797454, 4.62732332062340080010821326124, 5.35761959889671297849470314880, 5.48533769426723532315148175920, 6.38357226037683613251325512203, 6.60528638854424029309717881312, 7.17660430637350958581965305651, 7.19130116723388732080911721109, 7.70259492117001425523138851049, 7.915566344163055930494806306469, 8.628030587435983077100898783984, 9.165280137550611520540442374426, 9.788309088208788376460864511642, 10.02849723603225236203874858098
