| L(s) = 1 | + 3-s − 4·7-s + 9-s − 2·11-s + 13-s + 2·19-s − 4·21-s + 2·23-s + 27-s + 10·29-s + 4·31-s − 2·33-s − 6·37-s + 39-s − 6·41-s − 8·43-s − 12·47-s + 9·49-s − 14·53-s + 2·57-s + 6·59-s + 2·61-s − 4·63-s + 4·67-s + 2·69-s + 14·73-s + 8·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.458·19-s − 0.872·21-s + 0.417·23-s + 0.192·27-s + 1.85·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s + 0.160·39-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 9/7·49-s − 1.92·53-s + 0.264·57-s + 0.781·59-s + 0.256·61-s − 0.503·63-s + 0.488·67-s + 0.240·69-s + 1.63·73-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.52849752267592, −14.86638446699549, −14.18718119246940, −13.70262236201878, −13.24640251508412, −12.87306314528372, −12.23531615647794, −11.80037872757987, −10.99505719604315, −10.29644760402280, −9.982321579961734, −9.517200007865586, −8.871296771718725, −8.228041133313945, −7.918133781288574, −6.863081299487959, −6.665127837414482, −6.143498435847132, −5.066500217610349, −4.844528339122449, −3.691231246138084, −3.267722174632728, −2.850040606565135, −1.982304644879773, −0.9812496177003697, 0,
0.9812496177003697, 1.982304644879773, 2.850040606565135, 3.267722174632728, 3.691231246138084, 4.844528339122449, 5.066500217610349, 6.143498435847132, 6.665127837414482, 6.863081299487959, 7.918133781288574, 8.228041133313945, 8.871296771718725, 9.517200007865586, 9.982321579961734, 10.29644760402280, 10.99505719604315, 11.80037872757987, 12.23531615647794, 12.87306314528372, 13.24640251508412, 13.70262236201878, 14.18718119246940, 14.86638446699549, 15.52849752267592
