| L(s) = 1 | − 5-s + 4·11-s − 7·13-s − 6·17-s − 3·19-s + 2·23-s + 25-s + 2·29-s − 7·31-s − 7·37-s − 8·41-s + 5·43-s + 10·47-s + 8·53-s − 4·55-s + 10·59-s + 6·61-s + 7·65-s + 3·67-s − 15·73-s + 79-s + 8·83-s + 6·85-s + 2·89-s + 3·95-s + 10·97-s + 12·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s − 1.94·13-s − 1.45·17-s − 0.688·19-s + 0.417·23-s + 1/5·25-s + 0.371·29-s − 1.25·31-s − 1.15·37-s − 1.24·41-s + 0.762·43-s + 1.45·47-s + 1.09·53-s − 0.539·55-s + 1.30·59-s + 0.768·61-s + 0.868·65-s + 0.366·67-s − 1.75·73-s + 0.112·79-s + 0.878·83-s + 0.650·85-s + 0.211·89-s + 0.307·95-s + 1.01·97-s + 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174092802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174092802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−7.55311287889642733471956148514, −6.99679184231869983659422912483, −6.69154824683797695794043088102, −5.62404331405499590281321249462, −4.86468879922236599932568591959, −4.25685697256270409283139224249, −3.60849674689591306270191805968, −2.50351556936894653484911033972, −1.90478453972109855776462452143, −0.50421673678792372260582407097,
0.50421673678792372260582407097, 1.90478453972109855776462452143, 2.50351556936894653484911033972, 3.60849674689591306270191805968, 4.25685697256270409283139224249, 4.86468879922236599932568591959, 5.62404331405499590281321249462, 6.69154824683797695794043088102, 6.99679184231869983659422912483, 7.55311287889642733471956148514
