| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s − 2·9-s − 10-s − 12-s + 13-s + 14-s − 15-s + 16-s − 3·17-s + 2·18-s + 20-s + 21-s + 3·23-s + 24-s + 25-s − 26-s + 5·27-s − 28-s + 3·29-s + 30-s − 2·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 0.223·20-s + 0.218·21-s + 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.962·27-s − 0.188·28-s + 0.557·29-s + 0.182·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.387468145807354616552415073499, −7.41203911653452089631167682462, −6.57968796922367847067171292695, −6.13005038811746885179927108709, −5.35473782839236533164585565856, −4.45294298120265355043630575162, −3.19791918645303618709240478539, −2.44654156724732569306340487123, −1.21791060407224254343058550561, 0,
1.21791060407224254343058550561, 2.44654156724732569306340487123, 3.19791918645303618709240478539, 4.45294298120265355043630575162, 5.35473782839236533164585565856, 6.13005038811746885179927108709, 6.57968796922367847067171292695, 7.41203911653452089631167682462, 8.387468145807354616552415073499
