L(s) = 1 | + 2-s + 2.02·3-s + 4-s − 0.795·5-s + 2.02·6-s − 4.29·7-s + 8-s + 1.11·9-s − 0.795·10-s + 5.92·11-s + 2.02·12-s − 3.91·13-s − 4.29·14-s − 1.61·15-s + 16-s + 3.72·17-s + 1.11·18-s − 8.12·19-s − 0.795·20-s − 8.72·21-s + 5.92·22-s + 23-s + 2.02·24-s − 4.36·25-s − 3.91·26-s − 3.82·27-s − 4.29·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.17·3-s + 0.5·4-s − 0.355·5-s + 0.828·6-s − 1.62·7-s + 0.353·8-s + 0.372·9-s − 0.251·10-s + 1.78·11-s + 0.585·12-s − 1.08·13-s − 1.14·14-s − 0.416·15-s + 0.250·16-s + 0.903·17-s + 0.263·18-s − 1.86·19-s − 0.177·20-s − 1.90·21-s + 1.26·22-s + 0.208·23-s + 0.414·24-s − 0.873·25-s − 0.767·26-s − 0.735·27-s − 0.812·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 - 2.02T + 3T^{2} \) |
| 5 | \( 1 + 0.795T + 5T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 + 3.91T + 13T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 19 | \( 1 + 8.12T + 19T^{2} \) |
| 29 | \( 1 + 4.96T + 29T^{2} \) |
| 31 | \( 1 + 0.0401T + 31T^{2} \) |
| 37 | \( 1 - 0.836T + 37T^{2} \) |
| 41 | \( 1 - 2.26T + 41T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 - 0.452T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 - 1.58T + 61T^{2} \) |
| 67 | \( 1 + 4.00T + 67T^{2} \) |
| 71 | \( 1 + 8.40T + 71T^{2} \) |
| 73 | \( 1 + 0.555T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 0.959T + 83T^{2} \) |
| 89 | \( 1 - 3.11T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.67743152542568091581716394552, −6.88553867506418212732331340238, −6.41877768767938817648475551977, −5.71643230877386273737133270774, −4.48057784074779434976178970201, −3.76620434086011625282593980973, −3.43017965052344805428783322964, −2.59199317626151626697170617272, −1.73389774518231934379088664642, 0,
1.73389774518231934379088664642, 2.59199317626151626697170617272, 3.43017965052344805428783322964, 3.76620434086011625282593980973, 4.48057784074779434976178970201, 5.71643230877386273737133270774, 6.41877768767938817648475551977, 6.88553867506418212732331340238, 7.67743152542568091581716394552