| L(s) = 1 | − 2-s − 3-s + 4-s + 2.73·5-s + 6-s + 3.73·7-s − 8-s − 2·9-s − 2.73·10-s − 12-s + 3.46·13-s − 3.73·14-s − 2.73·15-s + 16-s − 0.535·17-s + 2·18-s + 19-s + 2.73·20-s − 3.73·21-s − 23-s + 24-s + 2.46·25-s − 3.46·26-s + 5·27-s + 3.73·28-s − 1.73·29-s + 2.73·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.22·5-s + 0.408·6-s + 1.41·7-s − 0.353·8-s − 0.666·9-s − 0.863·10-s − 0.288·12-s + 0.960·13-s − 0.997·14-s − 0.705·15-s + 0.250·16-s − 0.129·17-s + 0.471·18-s + 0.229·19-s + 0.610·20-s − 0.814·21-s − 0.208·23-s + 0.204·24-s + 0.492·25-s − 0.679·26-s + 0.962·27-s + 0.705·28-s − 0.321·29-s + 0.498·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.743848989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743848989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 0.535T + 17T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 0.464T + 37T^{2} \) |
| 41 | \( 1 + 7.26T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 - 5.53T + 47T^{2} \) |
| 53 | \( 1 - 3.92T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 4.73T + 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8.19T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 1.80T + 89T^{2} \) |
| 97 | \( 1 - 6.53T + 97T^{2} \) |
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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
−8.566396875021535060429392152187, −7.77227666676305595634903426783, −6.72618362907029438530955267240, −6.18145430533075254112641389814, −5.40494735038357138726525029594, −5.01021047976185066630510244435, −3.73179439196756753725609952680, −2.49165185844933903023332061351, −1.76092782896980712459720622103, −0.887839607975114701993999935570,
0.887839607975114701993999935570, 1.76092782896980712459720622103, 2.49165185844933903023332061351, 3.73179439196756753725609952680, 5.01021047976185066630510244435, 5.40494735038357138726525029594, 6.18145430533075254112641389814, 6.72618362907029438530955267240, 7.77227666676305595634903426783, 8.566396875021535060429392152187
