| L(s) = 1 | − 2-s − 5-s − 4·7-s + 8-s + 10-s + 3·11-s + 10·13-s + 4·14-s − 16-s − 5·19-s − 3·22-s − 9·23-s − 10·26-s + 10·31-s + 4·35-s + 37-s + 5·38-s − 40-s − 18·41-s + 16·43-s + 9·46-s + 3·47-s + 9·49-s − 3·53-s − 3·55-s − 4·56-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.904·11-s + 2.77·13-s + 1.06·14-s − 1/4·16-s − 1.14·19-s − 0.639·22-s − 1.87·23-s − 1.96·26-s + 1.79·31-s + 0.676·35-s + 0.164·37-s + 0.811·38-s − 0.158·40-s − 2.81·41-s + 2.43·43-s + 1.32·46-s + 0.437·47-s + 9/7·49-s − 0.412·53-s − 0.404·55-s − 0.534·56-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8713109207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8713109207\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.51709608347645260497798879788, −10.36783792608617175543158343543, −10.07168291364820221057791435278, −9.355064303432800977377051480626, −9.086365342128938118089509718943, −8.667198894367066604143885943952, −8.162098870193630395491173674520, −8.145008856847045863980666410414, −7.25796293471592569460819498124, −6.55956811450086999867761815620, −6.52039550257386792119845546424, −5.96620368532644603489391085255, −5.74506590043386110283293384039, −4.46457509955978068175015267251, −4.21609934038519061271371208433, −3.51645164313624607442848208960, −3.45949015715836793389873988116, −2.35611336468706306394553021057, −1.48341619547663454710902631830, −0.61817907725453485142459810440,
0.61817907725453485142459810440, 1.48341619547663454710902631830, 2.35611336468706306394553021057, 3.45949015715836793389873988116, 3.51645164313624607442848208960, 4.21609934038519061271371208433, 4.46457509955978068175015267251, 5.74506590043386110283293384039, 5.96620368532644603489391085255, 6.52039550257386792119845546424, 6.55956811450086999867761815620, 7.25796293471592569460819498124, 8.145008856847045863980666410414, 8.162098870193630395491173674520, 8.667198894367066604143885943952, 9.086365342128938118089509718943, 9.355064303432800977377051480626, 10.07168291364820221057791435278, 10.36783792608617175543158343543, 10.51709608347645260497798879788
