| L(s) = 1 | − 4·11-s + 14·19-s − 4·29-s − 10·31-s − 24·41-s + 5·49-s − 12·59-s − 26·61-s + 8·71-s + 16·79-s + 32·89-s − 18·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 3.21·19-s − 0.742·29-s − 1.79·31-s − 3.74·41-s + 5/7·49-s − 1.56·59-s − 3.32·61-s + 0.949·71-s + 1.80·79-s + 3.39·89-s − 1.72·109-s − 0.909·121-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 + 1.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376322931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376322931\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−9.331267016838338257057768453634, −9.022214605839184837002472240470, −9.007211530053825258165719737331, −8.038921367365909268691109007420, −7.83992718465271691318793047168, −7.57155353245598585353263728992, −7.29141881078908523702797220482, −6.72499552654162640202476555473, −6.29333735884585306146626941060, −5.72987882520013115043714649035, −5.28384867730018054443664407051, −4.97167549924180377533605525190, −4.96453258611044985750381604574, −3.88006302093910801752301067089, −3.44936287262918569944219182265, −3.20129740375275135172333770452, −2.70026893751221242204309481457, −1.78632465233074706665791127973, −1.51883889523385533590476900725, −0.43394465675900629350731996999,
0.43394465675900629350731996999, 1.51883889523385533590476900725, 1.78632465233074706665791127973, 2.70026893751221242204309481457, 3.20129740375275135172333770452, 3.44936287262918569944219182265, 3.88006302093910801752301067089, 4.96453258611044985750381604574, 4.97167549924180377533605525190, 5.28384867730018054443664407051, 5.72987882520013115043714649035, 6.29333735884585306146626941060, 6.72499552654162640202476555473, 7.29141881078908523702797220482, 7.57155353245598585353263728992, 7.83992718465271691318793047168, 8.038921367365909268691109007420, 9.007211530053825258165719737331, 9.022214605839184837002472240470, 9.331267016838338257057768453634
