| L(s) = 1 | + 2-s + 3-s + 4-s + 3.81·5-s + 6-s + 4.46·7-s + 8-s + 9-s + 3.81·10-s − 1.59·11-s + 12-s − 13-s + 4.46·14-s + 3.81·15-s + 16-s + 2.26·17-s + 18-s − 7.26·19-s + 3.81·20-s + 4.46·21-s − 1.59·22-s + 3.73·23-s + 24-s + 9.53·25-s − 26-s + 27-s + 4.46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.70·5-s + 0.408·6-s + 1.68·7-s + 0.353·8-s + 0.333·9-s + 1.20·10-s − 0.481·11-s + 0.288·12-s − 0.277·13-s + 1.19·14-s + 0.984·15-s + 0.250·16-s + 0.549·17-s + 0.235·18-s − 1.66·19-s + 0.852·20-s + 0.974·21-s − 0.340·22-s + 0.779·23-s + 0.204·24-s + 1.90·25-s − 0.196·26-s + 0.192·27-s + 0.844·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.129192279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.129192279\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 3.81T + 5T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + 7.26T + 19T^{2} \) |
| 23 | \( 1 - 3.73T + 23T^{2} \) |
| 29 | \( 1 + 0.246T + 29T^{2} \) |
| 31 | \( 1 + 4.75T + 31T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 + 4.71T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 - 7.94T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 - 7.06T + 67T^{2} \) |
| 71 | \( 1 - 1.51T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 7.04T + 83T^{2} \) |
| 89 | \( 1 + 3.12T + 89T^{2} \) |
| 97 | \( 1 + 8.69T + 97T^{2} \) |
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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
−7.992208470210673088506696623356, −6.94768028221814786807697072756, −6.45230814958090705710051103892, −5.41171656349016501451627294688, −5.17281920549997141161768732963, −4.49833180193584134748796266701, −3.47352068543287047136541370751, −2.38598784057078044256210701669, −2.04523865865996024758720697549, −1.31583749230059509228673796369,
1.31583749230059509228673796369, 2.04523865865996024758720697549, 2.38598784057078044256210701669, 3.47352068543287047136541370751, 4.49833180193584134748796266701, 5.17281920549997141161768732963, 5.41171656349016501451627294688, 6.45230814958090705710051103892, 6.94768028221814786807697072756, 7.992208470210673088506696623356
