| L(s) = 1 | − 2·4-s + 5.19·7-s + 4·16-s + 3.46·19-s − 5·25-s − 10.3·28-s − 8.66·31-s + 6.92·37-s + 13·43-s + 20·49-s + 13·61-s − 8·64-s − 12.1·67-s + 1.73·73-s − 6.92·76-s + 13·79-s + 19.0·97-s + 10·100-s + 13·103-s + 8.66·109-s + 20.7·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.96·7-s + 16-s + 0.794·19-s − 25-s − 1.96·28-s − 1.55·31-s + 1.13·37-s + 1.98·43-s + 2.85·49-s + 1.66·61-s − 64-s − 1.48·67-s + 0.202·73-s − 0.794·76-s + 1.46·79-s + 1.93·97-s + 100-s + 1.28·103-s + 0.829·109-s + 1.96·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.695954005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695954005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 5.19T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8.66T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 13T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−9.278550982516379096278423471452, −8.728135637863489422975384731820, −7.72536714583701354336606708079, −7.57777450618659838656119570441, −5.86759524499834832980391948758, −5.24169675575209242293511872037, −4.48531479108800978823334132249, −3.72835706033966006853332815739, −2.15113593829640636363996300451, −0.998216983262098798233120834662,
0.998216983262098798233120834662, 2.15113593829640636363996300451, 3.72835706033966006853332815739, 4.48531479108800978823334132249, 5.24169675575209242293511872037, 5.86759524499834832980391948758, 7.57777450618659838656119570441, 7.72536714583701354336606708079, 8.728135637863489422975384731820, 9.278550982516379096278423471452
