L(s) = 1 | + i·2-s + 2i·3-s − 4-s − 2·6-s − 4i·7-s − i·8-s − 9-s + 6·11-s − 2i·12-s − 2i·13-s + 4·14-s + 16-s − i·17-s − i·18-s + 4·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.15i·3-s − 0.5·4-s − 0.816·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s + 1.80·11-s − 0.577i·12-s − 0.554i·13-s + 1.06·14-s + 0.250·16-s − 0.242i·17-s − 0.235i·18-s + 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41834 + 0.876587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41834 + 0.876587i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−10.20807021126557963620939663778, −9.440113303225842820925469582869, −8.895507913691649505058478790362, −7.53053417169962023159770570643, −7.06334717884170801419828563191, −5.95213729209203949978645167188, −4.86017722052553836952731781614, −4.00020843161549923697352061725, −3.55695567842008071644403755670, −1.04461264161744860906713120235,
1.30615641770954580185602776096, 2.09534388736015774138814071507, 3.31148710761888674559528680209, 4.55757071986258563064423828608, 5.85664286162302742235136908323, 6.48577574925485111932275131191, 7.48574571426595025867574099021, 8.556654872303588707332457147324, 9.177562647460793801028747503700, 9.827723101829136760177926625586