L(s) = 1 | − 5-s − 11-s − 4·13-s − 7·17-s − 19-s + 23-s + 25-s − 3·29-s − 6·31-s − 3·37-s + 2·41-s − 7·43-s − 3·47-s − 10·53-s + 55-s + 59-s − 4·61-s + 4·65-s + 10·67-s − 3·71-s + 8·73-s − 4·79-s − 10·83-s + 7·85-s + 2·89-s + 95-s + 17·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.10·13-s − 1.69·17-s − 0.229·19-s + 0.208·23-s + 1/5·25-s − 0.557·29-s − 1.07·31-s − 0.493·37-s + 0.312·41-s − 1.06·43-s − 0.437·47-s − 1.37·53-s + 0.134·55-s + 0.130·59-s − 0.512·61-s + 0.496·65-s + 1.22·67-s − 0.356·71-s + 0.936·73-s − 0.450·79-s − 1.09·83-s + 0.759·85-s + 0.211·89-s + 0.102·95-s + 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−12.78387867812069, −12.27308417360882, −11.71089351941205, −11.30539961530071, −10.96521309067931, −10.51062119614549, −9.981327695964462, −9.456049882907511, −9.078681868112229, −8.653364563495207, −8.058746563695639, −7.685790653321622, −7.196170828914770, −6.620137338001332, −6.506870816482866, −5.576865099070724, −5.187911966851905, −4.731985439403680, −4.240828075424904, −3.753654686928291, −3.094921341935940, −2.606741167662405, −1.975784453185887, −1.596737558041670, −0.4685872012392072, 0,
0.4685872012392072, 1.596737558041670, 1.975784453185887, 2.606741167662405, 3.094921341935940, 3.753654686928291, 4.240828075424904, 4.731985439403680, 5.187911966851905, 5.576865099070724, 6.506870816482866, 6.620137338001332, 7.196170828914770, 7.685790653321622, 8.058746563695639, 8.653364563495207, 9.078681868112229, 9.456049882907511, 9.981327695964462, 10.51062119614549, 10.96521309067931, 11.30539961530071, 11.71089351941205, 12.27308417360882, 12.78387867812069