| L(s) = 1 | + 2-s + 3-s + 4-s − 2.59·5-s + 6-s − 1.23·7-s + 8-s + 9-s − 2.59·10-s − 0.622·11-s + 12-s − 1.32·13-s − 1.23·14-s − 2.59·15-s + 16-s − 4.70·17-s + 18-s + 3.79·19-s − 2.59·20-s − 1.23·21-s − 0.622·22-s + 24-s + 1.73·25-s − 1.32·26-s + 27-s − 1.23·28-s − 1.11·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.16·5-s + 0.408·6-s − 0.467·7-s + 0.353·8-s + 0.333·9-s − 0.820·10-s − 0.187·11-s + 0.288·12-s − 0.367·13-s − 0.330·14-s − 0.669·15-s + 0.250·16-s − 1.14·17-s + 0.235·18-s + 0.869·19-s − 0.580·20-s − 0.269·21-s − 0.132·22-s + 0.204·24-s + 0.346·25-s − 0.259·26-s + 0.192·27-s − 0.233·28-s − 0.206·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2.59T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 0.622T + 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 + 4.70T + 17T^{2} \) |
| 19 | \( 1 - 3.79T + 19T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 - 9.45T + 31T^{2} \) |
| 37 | \( 1 + 1.62T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 1.67T + 43T^{2} \) |
| 47 | \( 1 + 6.94T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 9.74T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 6.12T + 67T^{2} \) |
| 71 | \( 1 - 6.49T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.112536676277992457361384850858, −7.59046731046627892353365257110, −6.80786645849279009414889288684, −6.15730174929384296003870342668, −4.81434448280606088084068191448, −4.50237350425100032568564137249, −3.33454672441543310812298925233, −3.04193838409862620950700019671, −1.73002909981700688605930678584, 0,
1.73002909981700688605930678584, 3.04193838409862620950700019671, 3.33454672441543310812298925233, 4.50237350425100032568564137249, 4.81434448280606088084068191448, 6.15730174929384296003870342668, 6.80786645849279009414889288684, 7.59046731046627892353365257110, 8.112536676277992457361384850858
