L(s) = 1 | − 3-s + 5-s − 1.25i·7-s + 9-s − 0.630i·11-s − 3.96i·13-s − 15-s − 6.90·17-s + (2.11 + 3.81i)19-s + 1.25i·21-s + 6.49i·23-s + 25-s − 27-s + 2.83i·29-s + 7.48·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.473i·7-s + 0.333·9-s − 0.189i·11-s − 1.09i·13-s − 0.258·15-s − 1.67·17-s + (0.485 + 0.874i)19-s + 0.273i·21-s + 1.35i·23-s + 0.200·25-s − 0.192·27-s + 0.526i·29-s + 1.34·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.539796556\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.539796556\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + (-2.11 - 3.81i)T \) |
good | 7 | \( 1 + 1.25iT - 7T^{2} \) |
| 11 | \( 1 + 0.630iT - 11T^{2} \) |
| 13 | \( 1 + 3.96iT - 13T^{2} \) |
| 17 | \( 1 + 6.90T + 17T^{2} \) |
| 23 | \( 1 - 6.49iT - 23T^{2} \) |
| 29 | \( 1 - 2.83iT - 29T^{2} \) |
| 31 | \( 1 - 7.48T + 31T^{2} \) |
| 37 | \( 1 - 9.57iT - 37T^{2} \) |
| 41 | \( 1 + 7.74iT - 41T^{2} \) |
| 43 | \( 1 - 7.37iT - 43T^{2} \) |
| 47 | \( 1 - 3.33iT - 47T^{2} \) |
| 53 | \( 1 + 9.95iT - 53T^{2} \) |
| 59 | \( 1 - 9.92T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 7.82T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 - 7.64T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 9.45iT - 83T^{2} \) |
| 89 | \( 1 + 4.52iT - 89T^{2} \) |
| 97 | \( 1 - 3.90iT - 97T^{2} \) |
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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
−8.234557985534611547263375415444, −7.59881733329374227678668296868, −6.73075379523426691814363235877, −6.17991797577030426889883239685, −5.37281964820982210188537058197, −4.76992045095052666943212253306, −3.79389942318803830435738747078, −2.94637042352749046771741504652, −1.77723365497177063603607282720, −0.75478949611862918045126438782,
0.67147440344093626947873014202, 2.11709967254400859871954123100, 2.55486135917603424188996782986, 4.08353145136362992633932332640, 4.61725265701313061030402212483, 5.35525042499693529333910249899, 6.41795808464730972999552960424, 6.56175813565123375399988822031, 7.43651221636249336650937061446, 8.535360506338431587824758427095