L(s) = 1 | − 1.78·3-s − 0.701·5-s + 2.02·7-s + 0.187·9-s + 0.626·11-s − 2.93·13-s + 1.25·15-s − 2.71·17-s + 1.11·19-s − 3.62·21-s + 0.412·23-s − 4.50·25-s + 5.02·27-s + 6.46·29-s + 4.14·31-s − 1.11·33-s − 1.42·35-s + 2.98·37-s + 5.24·39-s − 0.135·41-s + 0.861·43-s − 0.131·45-s − 1.67·47-s − 2.87·49-s + 4.83·51-s − 8.51·53-s − 0.439·55-s + ⋯ |
L(s) = 1 | − 1.03·3-s − 0.313·5-s + 0.767·7-s + 0.0624·9-s + 0.189·11-s − 0.814·13-s + 0.323·15-s − 0.657·17-s + 0.254·19-s − 0.790·21-s + 0.0859·23-s − 0.901·25-s + 0.966·27-s + 1.20·29-s + 0.743·31-s − 0.194·33-s − 0.240·35-s + 0.490·37-s + 0.839·39-s − 0.0211·41-s + 0.131·43-s − 0.0195·45-s − 0.244·47-s − 0.411·49-s + 0.677·51-s − 1.16·53-s − 0.0592·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 5 | \( 1 + 0.701T + 5T^{2} \) |
| 7 | \( 1 - 2.02T + 7T^{2} \) |
| 11 | \( 1 - 0.626T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 + 2.71T + 17T^{2} \) |
| 19 | \( 1 - 1.11T + 19T^{2} \) |
| 23 | \( 1 - 0.412T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 - 4.14T + 31T^{2} \) |
| 37 | \( 1 - 2.98T + 37T^{2} \) |
| 41 | \( 1 + 0.135T + 41T^{2} \) |
| 43 | \( 1 - 0.861T + 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 + 8.51T + 53T^{2} \) |
| 59 | \( 1 - 0.406T + 59T^{2} \) |
| 61 | \( 1 + 8.53T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 2.78T + 71T^{2} \) |
| 73 | \( 1 + 2.11T + 73T^{2} \) |
| 79 | \( 1 - 0.317T + 79T^{2} \) |
| 83 | \( 1 - 5.05T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 7.68T + 97T^{2} \) |
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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
−7.56653041290657290379467637282, −6.54286363849779373403837562127, −6.25491585586998475988147168703, −5.21603687899626781668021443346, −4.82919088861749603138321487022, −4.18441552387584192800625452641, −3.06429794135468887247100271352, −2.16522293425519912462191933341, −1.05648492980120982710552705863, 0,
1.05648492980120982710552705863, 2.16522293425519912462191933341, 3.06429794135468887247100271352, 4.18441552387584192800625452641, 4.82919088861749603138321487022, 5.21603687899626781668021443346, 6.25491585586998475988147168703, 6.54286363849779373403837562127, 7.56653041290657290379467637282