| L(s) = 1 | − 2-s − 2·3-s + 4-s − 5-s + 2·6-s − 7-s − 8-s + 9-s + 10-s − 2·12-s − 2·13-s + 14-s + 2·15-s + 16-s − 3·17-s − 18-s + 7·19-s − 20-s + 2·21-s + 6·23-s + 2·24-s + 25-s + 2·26-s + 4·27-s − 28-s + 6·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s − 0.554·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 1.60·19-s − 0.223·20-s + 0.436·21-s + 1.25·23-s + 0.408·24-s + 1/5·25-s + 0.392·26-s + 0.769·27-s − 0.188·28-s + 1.11·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5867039282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5867039282\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.62045279265298829707306461064, −7.06867474509161455424923514102, −6.54941612682577203509194335097, −5.82605354451222235401745255543, −5.02406371372098052875761542205, −4.55669072155184862737782827726, −3.25099512621050457149790122193, −2.72745649244726764896137075460, −1.30882740033206838425424331747, −0.49018022649534537669397568133,
0.49018022649534537669397568133, 1.30882740033206838425424331747, 2.72745649244726764896137075460, 3.25099512621050457149790122193, 4.55669072155184862737782827726, 5.02406371372098052875761542205, 5.82605354451222235401745255543, 6.54941612682577203509194335097, 7.06867474509161455424923514102, 7.62045279265298829707306461064
