| L(s) = 1 | − 5-s − 3·7-s + 2·11-s + 2·13-s − 8·17-s + 16·19-s − 3·23-s − 29-s + 3·35-s − 8·37-s + 5·41-s − 8·43-s − 7·47-s + 7·49-s + 4·53-s − 2·55-s + 14·59-s − 7·61-s − 2·65-s − 3·67-s + 4·71-s + 8·73-s − 6·77-s − 6·79-s − 9·83-s + 8·85-s + 30·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 0.603·11-s + 0.554·13-s − 1.94·17-s + 3.67·19-s − 0.625·23-s − 0.185·29-s + 0.507·35-s − 1.31·37-s + 0.780·41-s − 1.21·43-s − 1.02·47-s + 49-s + 0.549·53-s − 0.269·55-s + 1.82·59-s − 0.896·61-s − 0.248·65-s − 0.366·67-s + 0.474·71-s + 0.936·73-s − 0.683·77-s − 0.675·79-s − 0.987·83-s + 0.867·85-s + 3.17·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.549606717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.549606717\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.182333552587743522879109340350, −8.972851750246153059371661478709, −8.637045339356351693429894147685, −7.999028363172932456784622756375, −7.79714666363497556063716786405, −7.15963232031711822304763004830, −6.86076558467772617829355139205, −6.80146954073009393084333723606, −6.09972126679515788254286482328, −5.76885403874693467414759487386, −5.33859822947697229457739418892, −4.85036539563301627219381364584, −4.41520278927015778674928862294, −3.76493951923230503918851462601, −3.39753157169236030375758788848, −3.30214926400492194579736855240, −2.53992094950022295776080539180, −1.90893111527218360871266075467, −1.18913361725541306557264354993, −0.48109018104607831584581149190,
0.48109018104607831584581149190, 1.18913361725541306557264354993, 1.90893111527218360871266075467, 2.53992094950022295776080539180, 3.30214926400492194579736855240, 3.39753157169236030375758788848, 3.76493951923230503918851462601, 4.41520278927015778674928862294, 4.85036539563301627219381364584, 5.33859822947697229457739418892, 5.76885403874693467414759487386, 6.09972126679515788254286482328, 6.80146954073009393084333723606, 6.86076558467772617829355139205, 7.15963232031711822304763004830, 7.79714666363497556063716786405, 7.999028363172932456784622756375, 8.637045339356351693429894147685, 8.972851750246153059371661478709, 9.182333552587743522879109340350
