| L(s) = 1 | + 5-s − 11-s + 2·13-s + 2·17-s − 4·19-s + 25-s + 2·29-s + 8·31-s − 10·37-s − 6·41-s − 4·43-s + 2·53-s − 55-s − 12·59-s − 14·61-s + 2·65-s + 12·67-s + 6·73-s − 16·79-s − 4·83-s + 2·85-s − 14·89-s − 4·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 1.64·37-s − 0.937·41-s − 0.609·43-s + 0.274·53-s − 0.134·55-s − 1.56·59-s − 1.79·61-s + 0.248·65-s + 1.46·67-s + 0.702·73-s − 1.80·79-s − 0.439·83-s + 0.216·85-s − 1.48·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.761382143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.761382143\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.22501298866962, −12.50706412414236, −12.22851802138490, −11.80655496821402, −11.08433240777968, −10.67146284730727, −10.36374885276771, −9.760681087699755, −9.441958431488501, −8.662351714354669, −8.323887040084125, −8.089836635504919, −7.148198271420597, −6.867754305739505, −6.275953577176848, −5.851658734395826, −5.292446206390828, −4.741852535619246, −4.274806160959615, −3.558919893627268, −3.025759691501899, −2.530379921649655, −1.673673076323817, −1.372037765476343, −0.3674231247776041,
0.3674231247776041, 1.372037765476343, 1.673673076323817, 2.530379921649655, 3.025759691501899, 3.558919893627268, 4.274806160959615, 4.741852535619246, 5.292446206390828, 5.851658734395826, 6.275953577176848, 6.867754305739505, 7.148198271420597, 8.089836635504919, 8.323887040084125, 8.662351714354669, 9.441958431488501, 9.760681087699755, 10.36374885276771, 10.67146284730727, 11.08433240777968, 11.80655496821402, 12.22851802138490, 12.50706412414236, 13.22501298866962
