L(s) = 1 | + 6·11-s + 13-s + 3·17-s − 4·19-s − 3·23-s − 3·29-s + 5·31-s − 10·37-s + 9·41-s + 43-s − 9·53-s − 9·59-s − 11·61-s + 4·67-s − 12·71-s + 10·73-s + 10·79-s − 9·83-s − 6·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 1.80·11-s + 0.277·13-s + 0.727·17-s − 0.917·19-s − 0.625·23-s − 0.557·29-s + 0.898·31-s − 1.64·37-s + 1.40·41-s + 0.152·43-s − 1.23·53-s − 1.17·59-s − 1.40·61-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 1.12·79-s − 0.987·83-s − 0.635·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.63018816340041, −12.71127395455063, −12.45524968499962, −12.10323625280471, −11.54163177583385, −11.07487015778697, −10.66367152927452, −10.04796711497772, −9.539997703422280, −9.206014880236006, −8.601116536442399, −8.299356602829301, −7.535003337002964, −7.211292426127658, −6.415153027424123, −6.225406699398781, −5.774070964143162, −4.948032353338426, −4.414100219144789, −3.961998793769946, −3.450456124114391, −2.899134278879581, −1.977473278972637, −1.559043003012465, −0.9224462354511134, 0,
0.9224462354511134, 1.559043003012465, 1.977473278972637, 2.899134278879581, 3.450456124114391, 3.961998793769946, 4.414100219144789, 4.948032353338426, 5.774070964143162, 6.225406699398781, 6.415153027424123, 7.211292426127658, 7.535003337002964, 8.299356602829301, 8.601116536442399, 9.206014880236006, 9.539997703422280, 10.04796711497772, 10.66367152927452, 11.07487015778697, 11.54163177583385, 12.10323625280471, 12.45524968499962, 12.71127395455063, 13.63018816340041