| L(s) = 1 | − 9-s − 6·11-s + 25-s + 4·29-s + 6·79-s + 81-s + 6·99-s + 2·109-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 9-s − 6·11-s + 25-s + 4·29-s + 6·79-s + 81-s + 6·99-s + 2·109-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9691634481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9691634481\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.12847634956544679466214847759, −5.74992068900548257676617220545, −5.70487914526630575482755047929, −5.48755149614774367319524501951, −5.47027143955413791704570723321, −5.09249014686402229310431398467, −5.00045654183884443996016025024, −4.93492800529279464181311647437, −4.72186899647891386296479018245, −4.51705138939921724877371951478, −4.24782872207413838987047141355, −4.17421640244165999271667986347, −3.36227566511469527229379191226, −3.35682207026901733452000509870, −3.24575881232710713749357959971, −2.94748474161607105176198663380, −2.80808094018709957033736870909, −2.74401668348127194542872686047, −2.43596447516594170361767467658, −2.08573898517493310998111227735, −2.02642972495275825516799328445, −1.90983710187228794544647101830, −0.805452926310872466956013981577, −0.797496923419593349704582391592, −0.54202867160820004478156585201,
0.54202867160820004478156585201, 0.797496923419593349704582391592, 0.805452926310872466956013981577, 1.90983710187228794544647101830, 2.02642972495275825516799328445, 2.08573898517493310998111227735, 2.43596447516594170361767467658, 2.74401668348127194542872686047, 2.80808094018709957033736870909, 2.94748474161607105176198663380, 3.24575881232710713749357959971, 3.35682207026901733452000509870, 3.36227566511469527229379191226, 4.17421640244165999271667986347, 4.24782872207413838987047141355, 4.51705138939921724877371951478, 4.72186899647891386296479018245, 4.93492800529279464181311647437, 5.00045654183884443996016025024, 5.09249014686402229310431398467, 5.47027143955413791704570723321, 5.48755149614774367319524501951, 5.70487914526630575482755047929, 5.74992068900548257676617220545, 6.12847634956544679466214847759
