| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 13-s + 14-s + 16-s + 6·17-s − 4·19-s + 6·23-s − 5·25-s + 26-s + 28-s + 8·31-s + 32-s + 6·34-s + 2·37-s − 4·38-s − 4·43-s + 6·46-s + 6·47-s + 49-s − 5·50-s + 52-s + 56-s + 6·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 1.25·23-s − 25-s + 0.196·26-s + 0.188·28-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s − 0.648·38-s − 0.609·43-s + 0.884·46-s + 0.875·47-s + 1/7·49-s − 0.707·50-s + 0.138·52-s + 0.133·56-s + 0.781·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.973327229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.973327229\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.448089795162708261938826389090, −8.408186985546070048356781615288, −7.78209670110472060321573804821, −6.86749409126732442342323738197, −6.02002624790821740098186487257, −5.25930237032666547409438421626, −4.38911319045089210033771368586, −3.48579012749559392932250457300, −2.47725820805684354198061161114, −1.18862924143215577465653274813,
1.18862924143215577465653274813, 2.47725820805684354198061161114, 3.48579012749559392932250457300, 4.38911319045089210033771368586, 5.25930237032666547409438421626, 6.02002624790821740098186487257, 6.86749409126732442342323738197, 7.78209670110472060321573804821, 8.408186985546070048356781615288, 9.448089795162708261938826389090
