| L(s) = 1 | − 2-s − 7·4-s + 5·5-s + 15·8-s − 5·10-s − 52·11-s − 22·13-s + 41·16-s − 14·17-s + 20·19-s − 35·20-s + 52·22-s + 168·23-s + 25·25-s + 22·26-s − 230·29-s + 288·31-s − 161·32-s + 14·34-s − 34·37-s − 20·38-s + 75·40-s + 122·41-s − 188·43-s + 364·44-s − 168·46-s + 256·47-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s + 0.447·5-s + 0.662·8-s − 0.158·10-s − 1.42·11-s − 0.469·13-s + 0.640·16-s − 0.199·17-s + 0.241·19-s − 0.391·20-s + 0.503·22-s + 1.52·23-s + 1/5·25-s + 0.165·26-s − 1.47·29-s + 1.66·31-s − 0.889·32-s + 0.0706·34-s − 0.151·37-s − 0.0853·38-s + 0.296·40-s + 0.464·41-s − 0.666·43-s + 1.24·44-s − 0.538·46-s + 0.794·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 - 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 230 T + p^{3} T^{2} \) |
| 31 | \( 1 - 288 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 122 T + p^{3} T^{2} \) |
| 43 | \( 1 + 188 T + p^{3} T^{2} \) |
| 47 | \( 1 - 256 T + p^{3} T^{2} \) |
| 53 | \( 1 - 338 T + p^{3} T^{2} \) |
| 59 | \( 1 - 100 T + p^{3} T^{2} \) |
| 61 | \( 1 + 742 T + p^{3} T^{2} \) |
| 67 | \( 1 + 84 T + p^{3} T^{2} \) |
| 71 | \( 1 - 328 T + p^{3} T^{2} \) |
| 73 | \( 1 - 38 T + p^{3} T^{2} \) |
| 79 | \( 1 + 240 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1212 T + p^{3} T^{2} \) |
| 89 | \( 1 - 330 T + p^{3} T^{2} \) |
| 97 | \( 1 + 866 T + p^{3} T^{2} \) |
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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
−8.345974586849892821780011271236, −7.68627341519673364025158190285, −6.94557884434733801175499562212, −5.72185873760274270909608910748, −5.12546453418247416996039637407, −4.45787170658013598385391916500, −3.22943392499193833299814832639, −2.32231289653404189504721090567, −1.03445051124400785558446183060, 0,
1.03445051124400785558446183060, 2.32231289653404189504721090567, 3.22943392499193833299814832639, 4.45787170658013598385391916500, 5.12546453418247416996039637407, 5.72185873760274270909608910748, 6.94557884434733801175499562212, 7.68627341519673364025158190285, 8.345974586849892821780011271236
