| L(s) = 1 | + 24·25-s + 64·47-s + 12·49-s − 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 24/5·25-s + 9.33·47-s + 12/7·49-s − 7.27·121-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 − 8·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.46123962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.46123962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 11 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 36 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 48 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 53 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - p T^{2} )^{8} \) |
| 89 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{4} \) |
| 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
−3.48751940187955588055555950360, −3.33942890109556469678197738080, −3.10772240877143530047062135313, −3.06416911732364035769728536637, −3.00933100975332601936715494727, −2.79767189411040286462902307975, −2.77068895540799566649481538220, −2.62823592158666419324610246775, −2.54740397589118820005923765389, −2.54227279420013564164168531129, −2.32779245465469100649060415756, −2.27695709997569167446503436784, −2.08289389267028083667379570066, −2.06657461874434545597828514865, −1.79534181303822149971383920451, −1.50599181665335712718699869826, −1.48989280873890279490553645739, −1.24352877729876606987643966411, −1.12767580719912969012009912035, −0.938751701066292678368803405720, −0.866072096041394515205549784064, −0.849021045927260880705946263087, −0.78535435817670845648564722556, −0.30772383878657698685021740017, −0.25943345441366760993840817510,
0.25943345441366760993840817510, 0.30772383878657698685021740017, 0.78535435817670845648564722556, 0.849021045927260880705946263087, 0.866072096041394515205549784064, 0.938751701066292678368803405720, 1.12767580719912969012009912035, 1.24352877729876606987643966411, 1.48989280873890279490553645739, 1.50599181665335712718699869826, 1.79534181303822149971383920451, 2.06657461874434545597828514865, 2.08289389267028083667379570066, 2.27695709997569167446503436784, 2.32779245465469100649060415756, 2.54227279420013564164168531129, 2.54740397589118820005923765389, 2.62823592158666419324610246775, 2.77068895540799566649481538220, 2.79767189411040286462902307975, 3.00933100975332601936715494727, 3.06416911732364035769728536637, 3.10772240877143530047062135313, 3.33942890109556469678197738080, 3.48751940187955588055555950360
Plot not available for L-functions of degree greater than 10.
