| L(s) = 1 | + 3·5-s + 11-s − 5·13-s − 2·17-s + 3·19-s − 9·23-s + 25-s + 2·31-s + 2·37-s + 3·41-s − 3·43-s − 14·47-s − 14·49-s + 8·53-s + 3·55-s − 6·59-s − 10·61-s − 15·65-s + 8·67-s + 4·71-s − 8·73-s − 6·79-s + 10·83-s − 6·85-s − 6·89-s + 9·95-s − 14·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.301·11-s − 1.38·13-s − 0.485·17-s + 0.688·19-s − 1.87·23-s + 1/5·25-s + 0.359·31-s + 0.328·37-s + 0.468·41-s − 0.457·43-s − 2.04·47-s − 2·49-s + 1.09·53-s + 0.404·55-s − 0.781·59-s − 1.28·61-s − 1.86·65-s + 0.977·67-s + 0.474·71-s − 0.936·73-s − 0.675·79-s + 1.09·83-s − 0.650·85-s − 0.635·89-s + 0.923·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95883264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95883264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 226 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.33849397646070578405045600755, −7.32840980739345383142059780203, −6.67928311159464640823568075984, −6.40915306892093620376161310444, −6.06905145268612668170482599964, −6.02797001173122175031913424464, −5.40011740132329526261619039897, −5.12356395251971359849833645317, −4.68458018744703465604536205040, −4.65508225634938805559071661399, −3.83941928754823959055717449194, −3.77149178407266903494579367208, −3.07776011557526384155893938070, −2.76859821503143241849474354074, −2.17474621682320982619609892867, −2.13603206834015227783898084385, −1.48816015749874533837855189620, −1.21362980246137482195601316949, 0, 0,
1.21362980246137482195601316949, 1.48816015749874533837855189620, 2.13603206834015227783898084385, 2.17474621682320982619609892867, 2.76859821503143241849474354074, 3.07776011557526384155893938070, 3.77149178407266903494579367208, 3.83941928754823959055717449194, 4.65508225634938805559071661399, 4.68458018744703465604536205040, 5.12356395251971359849833645317, 5.40011740132329526261619039897, 6.02797001173122175031913424464, 6.06905145268612668170482599964, 6.40915306892093620376161310444, 6.67928311159464640823568075984, 7.32840980739345383142059780203, 7.33849397646070578405045600755
