| L(s) = 1 | − 6·9-s + 14·25-s − 16·29-s − 12·37-s + 4·53-s + 9·81-s − 36·109-s − 64·113-s + 42·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2·9-s + 14/5·25-s − 2.97·29-s − 1.97·37-s + 0.549·53-s + 81-s − 3.44·109-s − 6.02·113-s + 3.81·121-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 + 2.15·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.331752923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331752923\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 59 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 71 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.05772586572976720707087208582, −5.89451473759576217582381851468, −5.80680563672161414389848414610, −5.38958533770439402399831138133, −5.37332256989891547002227182851, −5.28129245286230549199410996406, −5.03752271224157640770749155658, −4.83836265251116708220453262023, −4.45718911169348796258652559789, −4.38333646573713903179415817372, −4.06491429003184682575964771361, −3.73106232913595061590065024399, −3.49017777583057608784682024149, −3.43586504301211844815523621348, −3.37703116151361071978689788192, −2.76451731965301252315267983802, −2.70078447403784391963116920644, −2.50424197798056060662472958555, −2.46114496763073073520910122708, −1.76337066112040734718035421638, −1.75972543563717985049008098268, −1.27447708269193610663524363703, −1.17898659719933570447120187007, −0.37287371486604497287843479984, −0.31867576906660456768172386288,
0.31867576906660456768172386288, 0.37287371486604497287843479984, 1.17898659719933570447120187007, 1.27447708269193610663524363703, 1.75972543563717985049008098268, 1.76337066112040734718035421638, 2.46114496763073073520910122708, 2.50424197798056060662472958555, 2.70078447403784391963116920644, 2.76451731965301252315267983802, 3.37703116151361071978689788192, 3.43586504301211844815523621348, 3.49017777583057608784682024149, 3.73106232913595061590065024399, 4.06491429003184682575964771361, 4.38333646573713903179415817372, 4.45718911169348796258652559789, 4.83836265251116708220453262023, 5.03752271224157640770749155658, 5.28129245286230549199410996406, 5.37332256989891547002227182851, 5.38958533770439402399831138133, 5.80680563672161414389848414610, 5.89451473759576217582381851468, 6.05772586572976720707087208582
