| L(s) = 1 | − 2·4-s − 7·13-s + 4·16-s − 7·19-s − 5·25-s − 7·31-s − 37-s + 5·43-s + 14·52-s + 14·61-s − 8·64-s + 11·67-s − 7·73-s + 14·76-s − 13·79-s + 14·97-s + 10·100-s − 7·103-s + 17·109-s + ⋯ |
| L(s) = 1 | − 4-s − 1.94·13-s + 16-s − 1.60·19-s − 25-s − 1.25·31-s − 0.164·37-s + 0.762·43-s + 1.94·52-s + 1.79·61-s − 64-s + 1.34·67-s − 0.819·73-s + 1.60·76-s − 1.46·79-s + 1.42·97-s + 100-s − 0.689·103-s + 1.62·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.44340961493860792194432079598, −9.751017393898932451994893756079, −8.947372931910945992290137358883, −7.991372095607934979439494362982, −7.07511528958958291144252655588, −5.72591573809169746727814745787, −4.78354615730792598375845969221, −3.87851392965924754220515773132, −2.26817487861975221274698654515, 0,
2.26817487861975221274698654515, 3.87851392965924754220515773132, 4.78354615730792598375845969221, 5.72591573809169746727814745787, 7.07511528958958291144252655588, 7.991372095607934979439494362982, 8.947372931910945992290137358883, 9.751017393898932451994893756079, 10.44340961493860792194432079598
