L(s) = 1 | + 2.01i·2-s − i·3-s − 2.07·4-s + 2.01·6-s − 1.01i·7-s − 0.153i·8-s − 9-s + 4.75·11-s + 2.07i·12-s − 0.103i·13-s + 2.05·14-s − 3.84·16-s + 5.83i·17-s − 2.01i·18-s + 0.724·19-s + ⋯ |
L(s) = 1 | + 1.42i·2-s − 0.577i·3-s − 1.03·4-s + 0.824·6-s − 0.385i·7-s − 0.0541i·8-s − 0.333·9-s + 1.43·11-s + 0.599i·12-s − 0.0287i·13-s + 0.549·14-s − 0.960·16-s + 1.41i·17-s − 0.475i·18-s + 0.166·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.842890705\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.842890705\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 2.01iT - 2T^{2} \) |
| 7 | \( 1 + 1.01iT - 7T^{2} \) |
| 11 | \( 1 - 4.75T + 11T^{2} \) |
| 13 | \( 1 + 0.103iT - 13T^{2} \) |
| 17 | \( 1 - 5.83iT - 17T^{2} \) |
| 19 | \( 1 - 0.724T + 19T^{2} \) |
| 23 | \( 1 + 9.07iT - 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 - 1.06T + 31T^{2} \) |
| 37 | \( 1 + 4.02iT - 37T^{2} \) |
| 41 | \( 1 - 7.20T + 41T^{2} \) |
| 43 | \( 1 - 8.62iT - 43T^{2} \) |
| 47 | \( 1 - 8.19iT - 47T^{2} \) |
| 53 | \( 1 + 4.36iT - 53T^{2} \) |
| 59 | \( 1 - 4.91T + 59T^{2} \) |
| 61 | \( 1 - 6.96T + 61T^{2} \) |
| 67 | \( 1 - 9.91iT - 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 8.63iT - 73T^{2} \) |
| 79 | \( 1 + 2.48T + 79T^{2} \) |
| 83 | \( 1 - 4.24iT - 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 + 6.69iT - 97T^{2} \) |
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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
−8.916539910662673409589611298691, −8.505469281759571599595877301670, −7.72075578510887228914345212233, −6.93554336835637961425906757689, −6.32889152350805434593946173555, −5.91263429356089837216210925218, −4.61787101274520211749069353041, −3.96018982576065090004806938599, −2.43687507801925806742011346676, −1.04847052097090395091230611644,
0.881588497108119330516263791023, 2.05263220309808373504520191116, 3.11359544236190108690834639806, 3.76666222073788711143197245217, 4.63613796280613574309516929655, 5.55038397622286526137680539178, 6.64446093347621904570515253352, 7.49056675285161233758336494893, 8.886583320985587325172351336881, 9.197590769871001325102381127331