| L(s) = 1 | − 2-s + 4-s − 5-s + 1.84·7-s − 8-s + 10-s − 0.832·11-s + 3.21·13-s − 1.84·14-s + 16-s + 7.95·17-s + 4.20·19-s − 20-s + 0.832·22-s + 6.57·23-s + 25-s − 3.21·26-s + 1.84·28-s − 3.03·29-s − 4.73·31-s − 32-s − 7.95·34-s − 1.84·35-s − 37-s − 4.20·38-s + 40-s − 2.73·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.697·7-s − 0.353·8-s + 0.316·10-s − 0.251·11-s + 0.892·13-s − 0.493·14-s + 0.250·16-s + 1.92·17-s + 0.965·19-s − 0.223·20-s + 0.177·22-s + 1.37·23-s + 0.200·25-s − 0.631·26-s + 0.348·28-s − 0.564·29-s − 0.849·31-s − 0.176·32-s − 1.36·34-s − 0.311·35-s − 0.164·37-s − 0.682·38-s + 0.158·40-s − 0.426·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.523771213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.523771213\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 + 0.832T + 11T^{2} \) |
| 13 | \( 1 - 3.21T + 13T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 - 6.57T + 23T^{2} \) |
| 29 | \( 1 + 3.03T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 + 5.03T + 43T^{2} \) |
| 47 | \( 1 + 9.11T + 47T^{2} \) |
| 53 | \( 1 + 3.95T + 53T^{2} \) |
| 59 | \( 1 - 3.69T + 59T^{2} \) |
| 61 | \( 1 - 5.40T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 3.55T + 83T^{2} \) |
| 89 | \( 1 - 6.55T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.418082615902023579256466057974, −8.014895910044704534917665876377, −7.35652470237348134929491376897, −6.60441600159169941427044406779, −5.45296833222026870746612625622, −5.07612099289584404579755987640, −3.63064946609788112375752761359, −3.15747580856256755274545797900, −1.70365065409675702201901427759, −0.882877305450239840656309263477,
0.882877305450239840656309263477, 1.70365065409675702201901427759, 3.15747580856256755274545797900, 3.63064946609788112375752761359, 5.07612099289584404579755987640, 5.45296833222026870746612625622, 6.60441600159169941427044406779, 7.35652470237348134929491376897, 8.014895910044704534917665876377, 8.418082615902023579256466057974
