L(s) = 1 | + (−0.974 − 1.22i)3-s + (−0.623 − 0.781i)5-s + (0.433 + 0.900i)7-s + (−0.321 + 1.40i)9-s + (−0.347 + 1.52i)15-s + (0.678 − 1.40i)21-s + (−1.40 + 0.678i)23-s + (−0.222 + 0.974i)25-s + (0.626 − 0.301i)27-s + (1.62 + 0.781i)29-s + (0.433 − 0.900i)35-s + (0.277 + 0.347i)41-s + (−1.21 + 1.52i)43-s + (1.30 − 0.626i)45-s + (0.347 + 1.52i)47-s + ⋯ |
L(s) = 1 | + (−0.974 − 1.22i)3-s + (−0.623 − 0.781i)5-s + (0.433 + 0.900i)7-s + (−0.321 + 1.40i)9-s + (−0.347 + 1.52i)15-s + (0.678 − 1.40i)21-s + (−1.40 + 0.678i)23-s + (−0.222 + 0.974i)25-s + (0.626 − 0.301i)27-s + (1.62 + 0.781i)29-s + (0.433 − 0.900i)35-s + (0.277 + 0.347i)41-s + (−1.21 + 1.52i)43-s + (1.30 − 0.626i)45-s + (0.347 + 1.52i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6237609937\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6237609937\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.623 + 0.781i)T \) |
| 7 | \( 1 + (-0.433 - 0.900i)T \) |
good | 3 | \( 1 + (0.974 + 1.22i)T + (-0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 13 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1.40 - 0.678i)T + (0.623 - 0.781i)T^{2} \) |
| 29 | \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 41 | \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (1.21 - 1.52i)T + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.347 - 1.52i)T + (-0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 61 | \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-0.193 + 0.846i)T + (-0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.358785415650033458831048133640, −7.970628030916402508102586313574, −7.24904326011572256949487591963, −6.33005867373873858394755790568, −5.81903374297771173928634429675, −5.04221771838932022024704308408, −4.40113764083080741905629171400, −3.06476326096413830901028466821, −1.86851935791526856574090796957, −1.10618767698904458490661547983,
0.46711404657144153919783925067, 2.30300176695711612860100726426, 3.63588560807637010403818997973, 4.02872254450270482330203266653, 4.73542750242663310351275015107, 5.52025202040431117190535326329, 6.47699648102777439067894284836, 6.94040745041350949054795038920, 7.988419467422051236790682121149, 8.477169841537834399042195444902