| L(s) = 1 | + 4·9-s − 8·11-s − 20·25-s + 8·43-s + 20·49-s − 72·67-s + 16·81-s − 32·99-s − 88·107-s − 72·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 2.41·11-s − 4·25-s + 1.21·43-s + 20/7·49-s − 8.79·67-s + 16/9·81-s − 3.21·99-s − 8.50·107-s − 6.77·113-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5185081006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5185081006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 2 T^{2} - 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + 2 p T^{2} + 54 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 18 T^{2} + 230 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 28 T^{2} + 438 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 10 T^{2} + 558 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 48 T^{2} + 1550 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 84 T^{2} + 3350 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + p T^{2} )^{4}( 1 + 12 T + p T^{2} )^{4} \) |
| 41 | \( ( 1 - 124 T^{2} + 6870 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 148 T^{2} + 9558 T^{4} + 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 202 T^{2} + 16974 T^{4} - 202 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 90 T^{2} + 8438 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 322 T^{2} + 39678 T^{4} - 322 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 28 T^{2} - 8202 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.06594858555162748021967459100, −3.87976148524041179331797189745, −3.74501735554315474561083745800, −3.58182989154661580621739301888, −3.51891910345488087499565160681, −3.11621714676802546006591578660, −3.07747092958344827474732211429, −2.95474203149162840549468464929, −2.86912225834152456485115752751, −2.76681975915033887801425970004, −2.62832086398838312662574312944, −2.47215679099092822824836616081, −2.45441641792036635727389849491, −2.22913975547399337083830572838, −2.11442204673839169603991048679, −1.94195535812275414410839245022, −1.67733030888271715829305687021, −1.54904256103521386022507299193, −1.33343290103424234497555988979, −1.32799114581774983947108099899, −1.27157353002624975882173072120, −1.05268417019826288013721941562, −0.38939218076096442909064139470, −0.27698812663805802859685586463, −0.14934394835277832769409658483,
0.14934394835277832769409658483, 0.27698812663805802859685586463, 0.38939218076096442909064139470, 1.05268417019826288013721941562, 1.27157353002624975882173072120, 1.32799114581774983947108099899, 1.33343290103424234497555988979, 1.54904256103521386022507299193, 1.67733030888271715829305687021, 1.94195535812275414410839245022, 2.11442204673839169603991048679, 2.22913975547399337083830572838, 2.45441641792036635727389849491, 2.47215679099092822824836616081, 2.62832086398838312662574312944, 2.76681975915033887801425970004, 2.86912225834152456485115752751, 2.95474203149162840549468464929, 3.07747092958344827474732211429, 3.11621714676802546006591578660, 3.51891910345488087499565160681, 3.58182989154661580621739301888, 3.74501735554315474561083745800, 3.87976148524041179331797189745, 4.06594858555162748021967459100
Plot not available for L-functions of degree greater than 10.
