| L(s) = 1 | − 2-s + 3-s + 4-s − 0.732·5-s − 6-s − 7-s − 8-s + 9-s + 0.732·10-s + 5.19·11-s + 12-s + 14-s − 0.732·15-s + 16-s − 2.26·17-s − 18-s − 0.464·19-s − 0.732·20-s − 21-s − 5.19·22-s − 7.46·23-s − 24-s − 4.46·25-s + 27-s − 28-s − 3.53·29-s + 0.732·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.327·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.231·10-s + 1.56·11-s + 0.288·12-s + 0.267·14-s − 0.189·15-s + 0.250·16-s − 0.550·17-s − 0.235·18-s − 0.106·19-s − 0.163·20-s − 0.218·21-s − 1.10·22-s − 1.55·23-s − 0.204·24-s − 0.892·25-s + 0.192·27-s − 0.188·28-s − 0.656·29-s + 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.732T + 5T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 + 0.464T + 19T^{2} \) |
| 23 | \( 1 + 7.46T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 - 3.26T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 - 8.46T + 41T^{2} \) |
| 43 | \( 1 + 8.73T + 43T^{2} \) |
| 47 | \( 1 - 3.92T + 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 + 8.92T + 59T^{2} \) |
| 61 | \( 1 - 3.73T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 7.26T + 71T^{2} \) |
| 73 | \( 1 + 1.46T + 73T^{2} \) |
| 79 | \( 1 - 9T + 79T^{2} \) |
| 83 | \( 1 + 9.26T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.79595565510396281140199756205, −6.96762544505450822739438257732, −6.35236701969356359766549416081, −5.80034976908050005797436296857, −4.33311834499681691549442479500, −3.99913991592907148968147221830, −3.08765140932561698917316733758, −2.12704623228384191663301870260, −1.33033726759727835316665417796, 0,
1.33033726759727835316665417796, 2.12704623228384191663301870260, 3.08765140932561698917316733758, 3.99913991592907148968147221830, 4.33311834499681691549442479500, 5.80034976908050005797436296857, 6.35236701969356359766549416081, 6.96762544505450822739438257732, 7.79595565510396281140199756205
