L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 2·13-s + 15-s − 2·17-s − 4·19-s + 21-s + 8·23-s + 25-s + 27-s − 6·29-s + 8·31-s + 35-s − 2·37-s + 2·39-s − 2·41-s − 12·43-s + 45-s + 8·47-s + 49-s − 2·51-s + 6·53-s − 4·57-s − 4·59-s + 2·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.169·35-s − 0.328·37-s + 0.320·39-s − 0.312·41-s − 1.82·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.045998666\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.045998666\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 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 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.34570173459606, −12.66646746450792, −12.09902174774941, −11.71255890004117, −10.99952293742133, −10.70624311996661, −10.34473910886214, −9.639749902597082, −9.182591599451694, −8.818233083794554, −8.350629912976721, −7.963231017933770, −7.249695670069193, −6.774110392025458, −6.452987631510761, −5.735201842752365, −5.230863236730673, −4.661387818882859, −4.222061361671332, −3.545533735869777, −3.018661210548489, −2.430097862165877, −1.846232104533092, −1.316086492132648, −0.5438893935556393,
0.5438893935556393, 1.316086492132648, 1.846232104533092, 2.430097862165877, 3.018661210548489, 3.545533735869777, 4.222061361671332, 4.661387818882859, 5.230863236730673, 5.735201842752365, 6.452987631510761, 6.774110392025458, 7.249695670069193, 7.963231017933770, 8.350629912976721, 8.818233083794554, 9.182591599451694, 9.639749902597082, 10.34473910886214, 10.70624311996661, 10.99952293742133, 11.71255890004117, 12.09902174774941, 12.66646746450792, 13.34570173459606