L(s) = 1 | + 1.41·2-s − i·3-s + 2.00·4-s − i·5-s − 1.41i·6-s − 0.585·7-s + 2.82·8-s − 9-s − 1.41i·10-s − 4.82i·11-s − 2.00i·12-s + i·13-s − 0.828·14-s − 15-s + 4.00·16-s − 1.41·17-s + ⋯ |
L(s) = 1 | + 1.00·2-s − 0.577i·3-s + 1.00·4-s − 0.447i·5-s − 0.577i·6-s − 0.221·7-s + 1.00·8-s − 0.333·9-s − 0.447i·10-s − 1.45i·11-s − 0.577i·12-s + 0.277i·13-s − 0.221·14-s − 0.258·15-s + 1.00·16-s − 0.342·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.114565326\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.114565326\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 - iT \) |
good | 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 + 4.82iT - 11T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 0.242iT - 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 + 10.2iT - 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + 6.48iT - 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 - 4.82iT - 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 - 0.343iT - 53T^{2} \) |
| 59 | \( 1 - 3.17iT - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 - 1.17iT - 67T^{2} \) |
| 71 | \( 1 - 4.82T + 71T^{2} \) |
| 73 | \( 1 - 8.58T + 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 - 7.65iT - 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 - 6.72T + 97T^{2} \) |
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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
−9.100089526109472335180481161653, −8.279745097271503458386430128080, −7.54313293225158278404930462605, −6.53111915405996632815655851254, −6.00332504028517107001877758701, −5.17790722183384553129135132253, −4.14967293699980985124666823655, −3.21610395328258209668244637367, −2.23592910235349755036143940126, −0.871538998233061268934599877208,
1.79837612912412369149625355229, 2.93268023043620584814641290223, 3.67072693385860023285196093991, 4.80152857425384719392127801309, 5.15391830942896605490556932038, 6.47455221693267939520418297351, 6.93572540300142579304872767186, 7.80858676638947459315516554346, 8.941375719210057985775715721832, 9.897771754022871608683271297414