| L(s) = 1 | + 2·5-s − 7-s − 3·9-s + 11-s − 6·13-s − 2·17-s + 8·19-s − 25-s − 29-s − 4·31-s − 2·35-s + 2·37-s − 2·41-s + 4·43-s − 6·45-s − 4·47-s + 49-s + 10·53-s + 2·55-s − 12·59-s − 2·61-s + 3·63-s − 12·65-s − 12·67-s + 8·71-s − 10·73-s − 77-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s + 0.301·11-s − 1.66·13-s − 0.485·17-s + 1.83·19-s − 1/5·25-s − 0.185·29-s − 0.718·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 0.894·45-s − 0.583·47-s + 1/7·49-s + 1.37·53-s + 0.269·55-s − 1.56·59-s − 0.256·61-s + 0.377·63-s − 1.48·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122743096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122743096\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.51418393212524, −12.97353539434043, −12.37223021499495, −11.92649907021063, −11.59145175216790, −11.06038759980649, −10.33728143830817, −9.989536592138943, −9.496739612605774, −9.109885211577801, −8.814301694833391, −7.811589867949318, −7.578676659358434, −7.028468756069623, −6.398908014531165, −5.888006806906796, −5.408411429864297, −5.077221280239011, −4.379815266766368, −3.640146120644992, −2.929787588228768, −2.651759166678800, −1.964444456027828, −1.286038714003417, −0.3066719653938983,
0.3066719653938983, 1.286038714003417, 1.964444456027828, 2.651759166678800, 2.929787588228768, 3.640146120644992, 4.379815266766368, 5.077221280239011, 5.408411429864297, 5.888006806906796, 6.398908014531165, 7.028468756069623, 7.578676659358434, 7.811589867949318, 8.814301694833391, 9.109885211577801, 9.496739612605774, 9.989536592138943, 10.33728143830817, 11.06038759980649, 11.59145175216790, 11.92649907021063, 12.37223021499495, 12.97353539434043, 13.51418393212524
