| L(s) = 1 | + 6·5-s − 7-s − 6·11-s + 12·17-s + 7·19-s + 19·25-s + 5·31-s − 6·35-s − 37-s + 6·47-s − 6·49-s − 36·55-s − 3·67-s + 15·73-s + 6·77-s + 27·79-s − 12·83-s + 72·85-s − 12·89-s + 42·95-s + 6·101-s + 5·103-s − 12·107-s − 5·109-s − 12·113-s − 12·119-s + 13·121-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 0.377·7-s − 1.80·11-s + 2.91·17-s + 1.60·19-s + 19/5·25-s + 0.898·31-s − 1.01·35-s − 0.164·37-s + 0.875·47-s − 6/7·49-s − 4.85·55-s − 0.366·67-s + 1.75·73-s + 0.683·77-s + 3.03·79-s − 1.31·83-s + 7.80·85-s − 1.27·89-s + 4.30·95-s + 0.597·101-s + 0.492·103-s − 1.16·107-s − 0.478·109-s − 1.12·113-s − 1.10·119-s + 1.18·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.867709390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.867709390\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 27 T + 322 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00618018602297004553007880282, −9.837652035559882592675033631905, −9.484821166444071539078997288989, −9.299750802981559027408933590020, −8.513528867911589940564655701781, −8.013045436342747691905712330580, −7.69510935076730065069043461413, −7.33547611781743160015597808384, −6.60243585751128372839360272086, −6.24230118974771331470176126924, −5.71716745923910514169809733608, −5.48892943691590873712993826794, −5.14740552790729556275239509453, −4.95335560551707424016632813275, −3.71641947114840228992975792657, −3.19548452408356376078755964195, −2.69059908666722207005943690410, −2.38012340947232393790553933578, −1.46824724639271500189539877841, −1.00678612609309744881587395639,
1.00678612609309744881587395639, 1.46824724639271500189539877841, 2.38012340947232393790553933578, 2.69059908666722207005943690410, 3.19548452408356376078755964195, 3.71641947114840228992975792657, 4.95335560551707424016632813275, 5.14740552790729556275239509453, 5.48892943691590873712993826794, 5.71716745923910514169809733608, 6.24230118974771331470176126924, 6.60243585751128372839360272086, 7.33547611781743160015597808384, 7.69510935076730065069043461413, 8.013045436342747691905712330580, 8.513528867911589940564655701781, 9.299750802981559027408933590020, 9.484821166444071539078997288989, 9.837652035559882592675033631905, 10.00618018602297004553007880282
