| L(s) = 1 | + 2·3-s + 3·9-s − 6·11-s − 12·23-s − 10·25-s + 4·27-s + 16·31-s − 12·33-s + 4·37-s + 24·47-s + 49-s − 12·53-s + 16·67-s − 24·69-s + 12·71-s − 20·75-s + 5·81-s + 24·89-s + 32·93-s − 20·97-s − 18·99-s + 16·103-s + 8·111-s − 12·113-s + 25·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 1.80·11-s − 2.50·23-s − 2·25-s + 0.769·27-s + 2.87·31-s − 2.08·33-s + 0.657·37-s + 3.50·47-s + 1/7·49-s − 1.64·53-s + 1.95·67-s − 2.88·69-s + 1.42·71-s − 2.30·75-s + 5/9·81-s + 2.54·89-s + 3.31·93-s − 2.03·97-s − 1.80·99-s + 1.57·103-s + 0.759·111-s − 1.12·113-s + 2.27·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 853776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 853776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.367432883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.367432883\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.073258369451057604870193482042, −7.82135830629284694847339959443, −7.74489471466623660805940258686, −7.02496431224024673895143306242, −6.35275504498479924704280125506, −5.98540521798352690518559815387, −5.59741927027536741290705780820, −4.90298807829743408461258718076, −4.39050801784138122129668187119, −3.95298079878205574312837610470, −3.50211312607626107046853269991, −2.51993528714783510147091367087, −2.51752502799515976298852414682, −1.91622177654250912425514391632, −0.65589506685878581545962178161,
0.65589506685878581545962178161, 1.91622177654250912425514391632, 2.51752502799515976298852414682, 2.51993528714783510147091367087, 3.50211312607626107046853269991, 3.95298079878205574312837610470, 4.39050801784138122129668187119, 4.90298807829743408461258718076, 5.59741927027536741290705780820, 5.98540521798352690518559815387, 6.35275504498479924704280125506, 7.02496431224024673895143306242, 7.74489471466623660805940258686, 7.82135830629284694847339959443, 8.073258369451057604870193482042
