| L(s) = 1 | + 2·5-s + 4·11-s − 2·13-s + 2·17-s − 4·19-s + 8·23-s − 25-s + 6·29-s − 8·31-s − 6·37-s − 6·41-s − 4·43-s − 2·53-s + 8·55-s − 4·59-s − 2·61-s − 4·65-s + 4·67-s − 8·71-s − 10·73-s − 8·79-s + 4·83-s + 4·85-s − 6·89-s − 8·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s − 0.274·53-s + 1.07·55-s − 0.520·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.900·79-s + 0.439·83-s + 0.433·85-s − 0.635·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.35010784944987, −14.77967925598380, −14.48779470924861, −13.95084277729804, −13.32893938345160, −12.90206235670008, −12.24647676552354, −11.83796891213080, −11.16014286238017, −10.53048886419548, −10.09920302125357, −9.457051391888114, −8.981098314907705, −8.588953592735030, −7.739893314857653, −6.933456612438325, −6.712332005755285, −5.987622362296613, −5.342854853294549, −4.813799508868719, −4.053023083298428, −3.319037741242158, −2.644538648842359, −1.722651599218883, −1.300731686051283, 0,
1.300731686051283, 1.722651599218883, 2.644538648842359, 3.319037741242158, 4.053023083298428, 4.813799508868719, 5.342854853294549, 5.987622362296613, 6.712332005755285, 6.933456612438325, 7.739893314857653, 8.588953592735030, 8.981098314907705, 9.457051391888114, 10.09920302125357, 10.53048886419548, 11.16014286238017, 11.83796891213080, 12.24647676552354, 12.90206235670008, 13.32893938345160, 13.95084277729804, 14.48779470924861, 14.77967925598380, 15.35010784944987
