L(s) = 1 | + i·2-s − 2i·3-s − 4-s + 2·6-s + i·7-s − i·8-s − 9-s − 11-s + 2i·12-s + 4.74i·13-s − 14-s + 16-s + 4.74i·17-s − i·18-s − 4.74·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.15i·3-s − 0.5·4-s + 0.816·6-s + 0.377i·7-s − 0.353i·8-s − 0.333·9-s − 0.301·11-s + 0.577i·12-s + 1.31i·13-s − 0.267·14-s + 0.250·16-s + 1.15i·17-s − 0.235i·18-s − 1.08·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7056051959\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7056051959\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 13 | \( 1 - 4.74iT - 13T^{2} \) |
| 17 | \( 1 - 4.74iT - 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 4.74iT - 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + 10.7iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 6.74iT - 47T^{2} \) |
| 53 | \( 1 - 1.25iT - 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 0.744iT - 73T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 + 5.25iT - 97T^{2} \) |
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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
−8.309917158022010951820439088044, −7.30369480307926921434398109231, −6.82870969136238654070966718824, −6.24082234363718918471375299577, −5.55298358448071608497555592020, −4.46882249974186746421445810296, −3.79895415519716520753697695194, −2.27639078179161344787410657401, −1.74374082647104431666516671008, −0.21363676351226511801867451600,
1.18028391239245325744026782129, 2.58986421721644309344638509783, 3.34333860331378217525537882397, 4.01272007557177809033572670855, 4.97456301908890527920036818237, 5.24503460955240620443207605418, 6.41559817648139838775850524510, 7.44504159318687899310623499094, 8.141430710499727023075370090725, 8.948292807841186817195685135541