| L(s) = 1 | + 2-s + 2·4-s − 2·5-s − 4·7-s + 5·8-s − 2·10-s + 11-s + 2·13-s − 4·14-s + 5·16-s + 4·17-s − 4·20-s + 22-s + 8·23-s + 5·25-s + 2·26-s − 8·28-s − 6·29-s + 8·31-s + 10·32-s + 4·34-s + 8·35-s + 12·37-s − 10·40-s − 2·41-s + 2·44-s + 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 0.894·5-s − 1.51·7-s + 1.76·8-s − 0.632·10-s + 0.301·11-s + 0.554·13-s − 1.06·14-s + 5/4·16-s + 0.970·17-s − 0.894·20-s + 0.213·22-s + 1.66·23-s + 25-s + 0.392·26-s − 1.51·28-s − 1.11·29-s + 1.43·31-s + 1.76·32-s + 0.685·34-s + 1.35·35-s + 1.97·37-s − 1.58·40-s − 0.312·41-s + 0.301·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 793881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 793881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.348005451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.348005451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−10.26205031788557261912072099837, −10.22049985721243641980253240623, −9.466938437780251318043288735939, −9.132410686771526839332357109509, −8.727124020242786316614315104357, −7.88573859324328509506318435503, −7.71101318737073414285787821404, −7.42498722599783234180767426532, −6.73601811624836872724165904567, −6.56887328227304386772173689193, −6.07105715749106229903630370057, −5.64035159893455773232008609140, −4.74135430231099498418151920390, −4.68609387465894310633434542259, −3.87745152374594180562195107457, −3.57403440866074713131758552083, −2.87560877548079138827309625847, −2.73600333415952620808292653054, −1.49687613688693668414753015426, −0.856145449798869226071935477126,
0.856145449798869226071935477126, 1.49687613688693668414753015426, 2.73600333415952620808292653054, 2.87560877548079138827309625847, 3.57403440866074713131758552083, 3.87745152374594180562195107457, 4.68609387465894310633434542259, 4.74135430231099498418151920390, 5.64035159893455773232008609140, 6.07105715749106229903630370057, 6.56887328227304386772173689193, 6.73601811624836872724165904567, 7.42498722599783234180767426532, 7.71101318737073414285787821404, 7.88573859324328509506318435503, 8.727124020242786316614315104357, 9.132410686771526839332357109509, 9.466938437780251318043288735939, 10.22049985721243641980253240623, 10.26205031788557261912072099837
