| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 4·6-s + 3·9-s − 4·12-s − 4·16-s + 6·18-s − 2·19-s − 6·25-s − 4·27-s + 8·29-s − 18·31-s − 8·32-s + 6·36-s + 6·37-s − 4·38-s + 12·47-s + 8·48-s − 12·50-s + 24·53-s − 8·54-s + 4·57-s + 16·58-s + 24·59-s − 36·62-s − 8·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s + 9-s − 1.15·12-s − 16-s + 1.41·18-s − 0.458·19-s − 6/5·25-s − 0.769·27-s + 1.48·29-s − 3.23·31-s − 1.41·32-s + 36-s + 0.986·37-s − 0.648·38-s + 1.75·47-s + 1.15·48-s − 1.69·50-s + 3.29·53-s − 1.08·54-s + 0.529·57-s + 2.10·58-s + 3.12·59-s − 4.57·62-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.090109359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.090109359\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.828648151345596153736554101594, −8.281244559015482682332319229048, −7.29485403540511127428179965078, −7.29254795769706133610428023090, −6.78325537260034591368785398088, −6.01621490934601452163404663718, −5.84278690300589592864666069852, −5.40855511243246313730427714163, −5.03090203649817702025598064487, −4.18739539622885934722496613081, −4.04530586363603743226969074324, −3.51834980769699570316939757660, −2.47062022887811000287520417928, −2.04117357649968623475261976339, −0.69023867370329802034851223268,
0.69023867370329802034851223268, 2.04117357649968623475261976339, 2.47062022887811000287520417928, 3.51834980769699570316939757660, 4.04530586363603743226969074324, 4.18739539622885934722496613081, 5.03090203649817702025598064487, 5.40855511243246313730427714163, 5.84278690300589592864666069852, 6.01621490934601452163404663718, 6.78325537260034591368785398088, 7.29254795769706133610428023090, 7.29485403540511127428179965078, 8.281244559015482682332319229048, 8.828648151345596153736554101594
