| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 2·11-s + 5·16-s + 8·17-s − 8·19-s − 4·22-s − 4·23-s − 10·25-s − 8·31-s + 6·32-s + 16·34-s − 4·37-s − 16·38-s + 8·41-s − 20·43-s − 6·44-s − 8·46-s − 20·50-s − 4·53-s + 8·59-s − 16·62-s + 7·64-s + 24·68-s − 16·71-s − 8·73-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.603·11-s + 5/4·16-s + 1.94·17-s − 1.83·19-s − 0.852·22-s − 0.834·23-s − 2·25-s − 1.43·31-s + 1.06·32-s + 2.74·34-s − 0.657·37-s − 2.59·38-s + 1.24·41-s − 3.04·43-s − 0.904·44-s − 1.17·46-s − 2.82·50-s − 0.549·53-s + 1.04·59-s − 2.03·62-s + 7/8·64-s + 2.91·68-s − 1.89·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 28 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 180 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 192 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
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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
−7.29542208757960868442894179899, −7.28107369343615028110287089325, −6.62500836270285503258202784829, −6.48213971345871216746309460409, −5.92242279539087911123370123140, −5.75617596550174521241084519878, −5.38064030802644576590395019765, −5.35782790963189340005352511643, −4.62001275458236101101790794161, −4.34411041864422905157916193890, −4.08251615875410683537663548595, −3.54888994758577472104825514634, −3.43080771749806930262085326024, −2.98583232802086492150019843370, −2.36392775752372471971761567888, −2.15734777430987458665248668046, −1.50616075878121207401718628994, −1.42685369789567281596929366119, 0, 0,
1.42685369789567281596929366119, 1.50616075878121207401718628994, 2.15734777430987458665248668046, 2.36392775752372471971761567888, 2.98583232802086492150019843370, 3.43080771749806930262085326024, 3.54888994758577472104825514634, 4.08251615875410683537663548595, 4.34411041864422905157916193890, 4.62001275458236101101790794161, 5.35782790963189340005352511643, 5.38064030802644576590395019765, 5.75617596550174521241084519878, 5.92242279539087911123370123140, 6.48213971345871216746309460409, 6.62500836270285503258202784829, 7.28107369343615028110287089325, 7.29542208757960868442894179899
