| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 8-s − 3·9-s + 12-s + 3·16-s − 3·18-s + 13·19-s + 6·23-s + 24-s − 4·25-s + 4·27-s + 3·32-s + 3·36-s − 6·37-s + 13·38-s + 3·41-s + 6·46-s − 3·48-s − 49-s − 4·50-s + 4·54-s − 13·57-s − 17·61-s − 5·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.353·8-s − 9-s + 0.288·12-s + 3/4·16-s − 0.707·18-s + 2.98·19-s + 1.25·23-s + 0.204·24-s − 4/5·25-s + 0.769·27-s + 0.530·32-s + 1/2·36-s − 0.986·37-s + 2.10·38-s + 0.468·41-s + 0.884·46-s − 0.433·48-s − 1/7·49-s − 0.565·50-s + 0.544·54-s − 1.72·57-s − 2.17·61-s − 5/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394346 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394346 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.632395784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632395784\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 7 T + p T^{2} ) \) |
| 73 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−8.877138158879103659681076892803, −8.087785635283753567499403527681, −7.59342064983623174686754349685, −7.42511600390410125587361210614, −6.72369795752143691886044933552, −5.98815946080522635058800089725, −5.76680915485167472518108581786, −5.27137362701715745273215893231, −4.94888849357542036568799303220, −4.49843372419212924829576875110, −3.58246380990593374585114064964, −3.26014569386235684543905560058, −2.82479441600697525056608599738, −1.58658837797378256130982737558, −0.69492295149660450963549283238,
0.69492295149660450963549283238, 1.58658837797378256130982737558, 2.82479441600697525056608599738, 3.26014569386235684543905560058, 3.58246380990593374585114064964, 4.49843372419212924829576875110, 4.94888849357542036568799303220, 5.27137362701715745273215893231, 5.76680915485167472518108581786, 5.98815946080522635058800089725, 6.72369795752143691886044933552, 7.42511600390410125587361210614, 7.59342064983623174686754349685, 8.087785635283753567499403527681, 8.877138158879103659681076892803
