| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s + 8-s − 10-s + 4·13-s − 15-s − 16-s + 6·17-s + 4·19-s − 24-s − 4·26-s + 27-s − 12·29-s + 30-s + 4·31-s − 6·34-s − 2·37-s − 4·38-s − 4·39-s + 40-s + 12·41-s + 16·43-s + 12·47-s + 48-s − 6·51-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s + 0.353·8-s − 0.316·10-s + 1.10·13-s − 0.258·15-s − 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.204·24-s − 0.784·26-s + 0.192·27-s − 2.22·29-s + 0.182·30-s + 0.718·31-s − 1.02·34-s − 0.328·37-s − 0.648·38-s − 0.640·39-s + 0.158·40-s + 1.87·41-s + 2.43·43-s + 1.75·47-s + 0.144·48-s − 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.697910482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.697910482\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.544787395011156506717450599516, −9.232894149250039570294093151057, −9.120917774752511047662408246777, −8.673005871746150827978191109198, −7.908547669625469590777683335485, −7.80287190912913755425123588092, −7.31557218086267308855342709734, −7.15801515094960464533727461500, −6.13219431412373771440099111812, −6.08259412524113675974725328484, −5.66832432164925006798603081972, −5.40704564487527155909722844026, −4.68066834888700370591490358587, −4.25332277444785585055789756385, −3.51398528563802691072863699888, −3.41083677419244620393415635545, −2.44407716191140478507327963006, −1.95968904583830715558699823249, −0.973447919990051242184857322500, −0.841438840184345022624982477454,
0.841438840184345022624982477454, 0.973447919990051242184857322500, 1.95968904583830715558699823249, 2.44407716191140478507327963006, 3.41083677419244620393415635545, 3.51398528563802691072863699888, 4.25332277444785585055789756385, 4.68066834888700370591490358587, 5.40704564487527155909722844026, 5.66832432164925006798603081972, 6.08259412524113675974725328484, 6.13219431412373771440099111812, 7.15801515094960464533727461500, 7.31557218086267308855342709734, 7.80287190912913755425123588092, 7.908547669625469590777683335485, 8.673005871746150827978191109198, 9.120917774752511047662408246777, 9.232894149250039570294093151057, 9.544787395011156506717450599516
