L(s) = 1 | + (−0.826 + 0.563i)4-s + (−0.955 + 0.294i)7-s + (1.06 − 1.14i)13-s + (0.365 − 0.930i)16-s + 1.94i·19-s + (−0.733 + 0.680i)25-s + (0.623 − 0.781i)28-s + (0.751 + 0.433i)31-s + (−1.62 + 0.781i)37-s + (0.162 − 0.414i)43-s + (0.826 − 0.563i)49-s + (−0.233 + 1.54i)52-s + (−0.488 + 0.716i)61-s + (0.222 + 0.974i)64-s + (−0.623 + 1.07i)67-s + ⋯ |
L(s) = 1 | + (−0.826 + 0.563i)4-s + (−0.955 + 0.294i)7-s + (1.06 − 1.14i)13-s + (0.365 − 0.930i)16-s + 1.94i·19-s + (−0.733 + 0.680i)25-s + (0.623 − 0.781i)28-s + (0.751 + 0.433i)31-s + (−1.62 + 0.781i)37-s + (0.162 − 0.414i)43-s + (0.826 − 0.563i)49-s + (−0.233 + 1.54i)52-s + (−0.488 + 0.716i)61-s + (0.222 + 0.974i)64-s + (−0.623 + 1.07i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6416787420\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6416787420\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.955 - 0.294i)T \) |
good | 2 | \( 1 + (0.826 - 0.563i)T^{2} \) |
| 5 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 11 | \( 1 + (0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (-1.06 + 1.14i)T + (-0.0747 - 0.997i)T^{2} \) |
| 17 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 - 1.94iT - T^{2} \) |
| 23 | \( 1 + (0.365 - 0.930i)T^{2} \) |
| 29 | \( 1 + (0.365 + 0.930i)T^{2} \) |
| 31 | \( 1 + (-0.751 - 0.433i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 41 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 43 | \( 1 + (-0.162 + 0.414i)T + (-0.733 - 0.680i)T^{2} \) |
| 47 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 53 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 61 | \( 1 + (0.488 - 0.716i)T + (-0.365 - 0.930i)T^{2} \) |
| 67 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (0.846 - 0.193i)T + (0.900 - 0.433i)T^{2} \) |
| 79 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 89 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 97 | \( 1 + (1.68 - 0.974i)T + (0.5 - 0.866i)T^{2} \) |
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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
−8.681656951964383069997229645418, −8.301640750561655982188785407539, −7.59226968167809478180585901925, −6.64072591538172823591654787391, −5.74227459904348139491474739621, −5.36513338838092815907224277861, −4.03370853233687819902553242732, −3.55230556418608911108931346684, −2.87036934236941822547148360105, −1.31842647763560610763178888034,
0.39860510380244530895529165259, 1.72189017989836671064417184500, 2.97515063583304998174439084369, 3.96857681654236469610954597660, 4.47574037842102275984902667852, 5.40400741698655495493659154660, 6.36637436876322281362007716241, 6.62857447536710351466643547776, 7.66702296599051633927881291135, 8.718090302485110364383064615981