| L(s) = 1 | − 7-s + 11-s + 6·13-s + 2·17-s − 6·23-s − 6·29-s + 2·31-s − 10·37-s + 8·41-s − 8·43-s − 4·47-s + 49-s + 6·53-s − 6·59-s − 8·61-s + 14·67-s − 8·71-s + 2·73-s − 77-s + 16·79-s − 12·83-s + 18·89-s − 6·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s + 1.66·13-s + 0.485·17-s − 1.25·23-s − 1.11·29-s + 0.359·31-s − 1.64·37-s + 1.24·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.781·59-s − 1.02·61-s + 1.71·67-s − 0.949·71-s + 0.234·73-s − 0.113·77-s + 1.80·79-s − 1.31·83-s + 1.90·89-s − 0.628·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.157102092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.157102092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−12.80675263305106, −12.30243306516791, −11.75486166825720, −11.48916244071283, −10.89340127247121, −10.42708988716970, −10.09740494308248, −9.467876979717657, −9.061078617696199, −8.596887930996108, −8.145802850925858, −7.652416757351861, −7.146297836017128, −6.473645508867518, −6.160253238049061, −5.778724426798759, −5.163944552885789, −4.578289570282991, −3.821084367359569, −3.602806083320879, −3.189192820908343, −2.245505565534491, −1.764421424838091, −1.162128144621085, −0.4141437327168005,
0.4141437327168005, 1.162128144621085, 1.764421424838091, 2.245505565534491, 3.189192820908343, 3.602806083320879, 3.821084367359569, 4.578289570282991, 5.163944552885789, 5.778724426798759, 6.160253238049061, 6.473645508867518, 7.146297836017128, 7.652416757351861, 8.145802850925858, 8.596887930996108, 9.061078617696199, 9.467876979717657, 10.09740494308248, 10.42708988716970, 10.89340127247121, 11.48916244071283, 11.75486166825720, 12.30243306516791, 12.80675263305106
