L(s) = 1 | + 1.41·2-s + 2.00·4-s + 2.23i·5-s + 1.52i·7-s + 2.82·8-s + 3.16i·10-s + 2.20i·11-s − 1.82·13-s + 2.15i·14-s + 4.00·16-s − 23.2i·17-s − 2.85i·19-s + 4.47i·20-s + 3.11i·22-s + (14.5 + 17.8i)23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.447i·5-s + 0.217i·7-s + 0.353·8-s + 0.316i·10-s + 0.200i·11-s − 0.140·13-s + 0.153i·14-s + 0.250·16-s − 1.36i·17-s − 0.150i·19-s + 0.223i·20-s + 0.141i·22-s + (0.632 + 0.774i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.360062882\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.360062882\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-14.5 - 17.8i)T \) |
good | 7 | \( 1 - 1.52iT - 49T^{2} \) |
| 11 | \( 1 - 2.20iT - 121T^{2} \) |
| 13 | \( 1 + 1.82T + 169T^{2} \) |
| 17 | \( 1 + 23.2iT - 289T^{2} \) |
| 19 | \( 1 + 2.85iT - 361T^{2} \) |
| 29 | \( 1 - 7.23T + 841T^{2} \) |
| 31 | \( 1 - 49.9T + 961T^{2} \) |
| 37 | \( 1 + 9.68iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 62.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 76.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 17.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 33.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 35.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 8.54iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 78.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 111.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 134.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 72.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 119. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 73.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 178. iT - 9.40e3T^{2} \) |
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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
−9.193419904064995028309290597353, −8.018485676891767872141593513861, −7.33979812849822072064564658943, −6.62895560268798149225464119122, −5.81279791820804674520387508015, −4.94607672169245824398853639234, −4.23497449095767592509929749690, −3.01329134590585294332252224486, −2.52706999993012740942746197743, −1.04210662387215495190721703657,
0.77475314837288556404821062242, 1.98580586074048368165039041789, 3.07487147813653391929538559759, 4.07929849263194484058962378450, 4.68941912223138971813898830880, 5.67612164515767497449993297438, 6.33939373629315418518664489443, 7.16962564763110842577654904354, 8.144703092676771866873439375996, 8.660707662060839105563364311538