| L(s) = 1 | − 2·4-s − 7-s + 11-s − 3·13-s + 4·16-s + 6·17-s − 23-s − 5·25-s + 2·28-s + 4·29-s − 4·31-s − 7·37-s + 3·41-s + 6·43-s − 2·44-s + 4·47-s + 49-s + 6·52-s + 6·53-s + 4·59-s + 2·61-s − 8·64-s + 3·67-s − 12·68-s − 13·71-s − 73-s − 77-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 0.301·11-s − 0.832·13-s + 16-s + 1.45·17-s − 0.208·23-s − 25-s + 0.377·28-s + 0.742·29-s − 0.718·31-s − 1.15·37-s + 0.468·41-s + 0.914·43-s − 0.301·44-s + 0.583·47-s + 1/7·49-s + 0.832·52-s + 0.824·53-s + 0.520·59-s + 0.256·61-s − 64-s + 0.366·67-s − 1.45·68-s − 1.54·71-s − 0.117·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.03384542384903, −12.59583020749624, −12.24591836197099, −11.82358660829724, −11.39434654036421, −10.44748048698972, −10.23732732787233, −9.909741303733783, −9.357761988463407, −8.908679144909912, −8.557591300419043, −7.860072040322619, −7.411706817710582, −7.214028675441565, −6.205446844421073, −5.949706509185289, −5.325604801296750, −4.998400517273906, −4.309084162522832, −3.798508260572819, −3.454080487909627, −2.760965013348391, −2.119103384639486, −1.316822996157696, −0.7107242744536768, 0,
0.7107242744536768, 1.316822996157696, 2.119103384639486, 2.760965013348391, 3.454080487909627, 3.798508260572819, 4.309084162522832, 4.998400517273906, 5.325604801296750, 5.949706509185289, 6.205446844421073, 7.214028675441565, 7.411706817710582, 7.860072040322619, 8.557591300419043, 8.908679144909912, 9.357761988463407, 9.909741303733783, 10.23732732787233, 10.44748048698972, 11.39434654036421, 11.82358660829724, 12.24591836197099, 12.59583020749624, 13.03384542384903
