| L(s) = 1 | − i·2-s − 3i·3-s + 3·4-s − 3·6-s − 7i·8-s − 9·9-s − 9i·12-s + 5·16-s + 14i·17-s + 9i·18-s + 22·19-s + 34i·23-s − 21·24-s + 27i·27-s + 2·31-s − 33i·32-s + ⋯ |
| L(s) = 1 | − 0.5i·2-s − i·3-s + 0.750·4-s − 0.5·6-s − 0.875i·8-s − 9-s − 0.750i·12-s + 0.312·16-s + 0.823i·17-s + 0.5i·18-s + 1.15·19-s + 1.47i·23-s − 0.875·24-s + i·27-s + 0.0645·31-s − 1.03i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.00795 - 1.00795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00795 - 1.00795i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + iT - 4T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 14iT - 289T^{2} \) |
| 19 | \( 1 - 22T + 361T^{2} \) |
| 23 | \( 1 - 34iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 2T + 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 14iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 86iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 118T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 + 98T + 6.24e3T^{2} \) |
| 83 | \( 1 - 154iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−13.76174265293518404568626967360, −12.76615296458415609702033283717, −11.83627455434379745526853383829, −11.04733130159140469325977840695, −9.638918405149932052416966794377, −7.995983009523014210497531158432, −6.98511199075596122388710643050, −5.75659578705248689833887837137, −3.23820338775733608420604808770, −1.57143054738411721911180195206,
2.92802658174998313751160576561, 4.84321135567696772073107231735, 6.12509675385144356007235067614, 7.55021665442649609824048918315, 8.879540449514900323300420046373, 10.15435910495107794465930274154, 11.18099988496288200452632945387, 12.11488173386944239510857561232, 13.92539311789386162852678130969, 14.76490280518785353993399146950
