L(s) = 1 | + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (−1.40 + 0.678i)5-s + (−0.900 − 0.433i)7-s + (0.433 + 0.900i)8-s + (0.678 − 1.40i)10-s + (0.347 − 0.277i)11-s + (0.974 − 0.222i)14-s + (−0.900 − 0.433i)16-s + (0.347 + 1.52i)20-s + (−0.0990 + 0.433i)22-s + (0.900 − 1.12i)25-s + (−0.623 + 0.781i)28-s + (−1.21 + 0.277i)29-s − 0.867i·31-s + (0.974 − 0.222i)32-s + ⋯ |
L(s) = 1 | + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (−1.40 + 0.678i)5-s + (−0.900 − 0.433i)7-s + (0.433 + 0.900i)8-s + (0.678 − 1.40i)10-s + (0.347 − 0.277i)11-s + (0.974 − 0.222i)14-s + (−0.900 − 0.433i)16-s + (0.347 + 1.52i)20-s + (−0.0990 + 0.433i)22-s + (0.900 − 1.12i)25-s + (−0.623 + 0.781i)28-s + (−1.21 + 0.277i)29-s − 0.867i·31-s + (0.974 − 0.222i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4742249905\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4742249905\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.781 - 0.623i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.900 + 0.433i)T \) |
good | 5 | \( 1 + (1.40 - 0.678i)T + (0.623 - 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.347 + 0.277i)T + (0.222 - 0.974i)T^{2} \) |
| 13 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 29 | \( 1 + (1.21 - 0.277i)T + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + 0.867iT - T^{2} \) |
| 37 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 53 | \( 1 + (-1.94 - 0.445i)T + (0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 61 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.678 - 0.541i)T + (0.222 + 0.974i)T^{2} \) |
| 79 | \( 1 - 1.80T + T^{2} \) |
| 83 | \( 1 + (-0.974 + 1.22i)T + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 97 | \( 1 - 1.94iT - T^{2} \) |
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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
−8.879384778059058920110103160695, −7.83206411044635047330219830467, −7.55697892831009952740647999478, −6.77122255061205671008319855214, −6.26488103932493252735498618713, −5.26493732095198287102123288131, −4.06794404839596328809577117433, −3.54356322843425252632292560453, −2.39763386975850100407715368784, −0.72888340095803033666342322758,
0.58846881001718281113149419803, 1.96946734406732853830004957826, 3.16310653971190927374045087929, 3.77672575523029970380489125305, 4.48607735871400219788875073179, 5.59627319957691446897656987874, 6.78257026843064695133570836429, 7.27817616932351897613915004448, 8.126884689431704610732702303612, 8.655821527550479670971728946788