| L(s) = 1 | + 4-s + 6·13-s − 2·25-s + 2·49-s + 6·52-s − 2·61-s − 64-s − 2·100-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 19·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + ⋯ |
| L(s) = 1 | + 4-s + 6·13-s − 2·25-s + 2·49-s + 6·52-s − 2·61-s − 64-s − 2·100-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 19·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.270874079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.270874079\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.87417665803887921583664907984, −6.47476685328454848412994025492, −6.30688976375522029569711531400, −6.22031393266451291824868806856, −6.13121024246363087341627813574, −6.08739896143301286395519270466, −5.65183261391244173302746590622, −5.53330195015241625009453675158, −5.43376046133416910566317264553, −4.84190186407333219080623164658, −4.82583962992809025588495286626, −4.27225547582750772383416098145, −4.02082850120711028131731895036, −3.88156644436308382415282703005, −3.80705727403318429549802322263, −3.65682689132238934106769251496, −3.39502238818422010992063958967, −2.99898421438659897999071535336, −2.63611711717537205240388623621, −2.59586186248190910979691187596, −2.06115072009436680961967416694, −1.64338055233249409097346739135, −1.44137647800351894646726898792, −1.32001599010555348249834510369, −0.984487932477870132049275227010,
0.984487932477870132049275227010, 1.32001599010555348249834510369, 1.44137647800351894646726898792, 1.64338055233249409097346739135, 2.06115072009436680961967416694, 2.59586186248190910979691187596, 2.63611711717537205240388623621, 2.99898421438659897999071535336, 3.39502238818422010992063958967, 3.65682689132238934106769251496, 3.80705727403318429549802322263, 3.88156644436308382415282703005, 4.02082850120711028131731895036, 4.27225547582750772383416098145, 4.82583962992809025588495286626, 4.84190186407333219080623164658, 5.43376046133416910566317264553, 5.53330195015241625009453675158, 5.65183261391244173302746590622, 6.08739896143301286395519270466, 6.13121024246363087341627813574, 6.22031393266451291824868806856, 6.30688976375522029569711531400, 6.47476685328454848412994025492, 6.87417665803887921583664907984
