| L(s) = 1 | − 4-s − 13-s − 4·19-s − 25-s − 31-s − 2·37-s + 43-s + 52-s − 61-s + 64-s + 67-s − 4·73-s + 4·76-s + 79-s − 97-s + 100-s − 103-s − 2·109-s − 121-s + 124-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s − 13-s − 4·19-s − 25-s − 31-s − 2·37-s + 43-s + 52-s − 61-s + 64-s + 67-s − 4·73-s + 4·76-s + 79-s − 97-s + 100-s − 103-s − 2·109-s − 121-s + 124-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06093800082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06093800082\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 + T )^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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
−8.877571291584862923906750319945, −8.392824157584411555567446582993, −8.379735428483331334628078498891, −7.64299557900313420112711155518, −7.52368204058708200658033409986, −6.83595776920209713632050686430, −6.71145171244274323641917800639, −6.25598652396076820376440431878, −5.85181350750069429419487708468, −5.39057015098913410445373539248, −5.07500998193578591289106920439, −4.46037453010578838859095741106, −4.38469397678215260440292043519, −3.81715086233612901347353920020, −3.80150031408423911583567271919, −2.74127038229931267554903226479, −2.53073662384467587119016249219, −1.87523129354449712348721903555, −1.61631726268690119997053072864, −0.13116800450612760535492710482,
0.13116800450612760535492710482, 1.61631726268690119997053072864, 1.87523129354449712348721903555, 2.53073662384467587119016249219, 2.74127038229931267554903226479, 3.80150031408423911583567271919, 3.81715086233612901347353920020, 4.38469397678215260440292043519, 4.46037453010578838859095741106, 5.07500998193578591289106920439, 5.39057015098913410445373539248, 5.85181350750069429419487708468, 6.25598652396076820376440431878, 6.71145171244274323641917800639, 6.83595776920209713632050686430, 7.52368204058708200658033409986, 7.64299557900313420112711155518, 8.379735428483331334628078498891, 8.392824157584411555567446582993, 8.877571291584862923906750319945
