| L(s) = 1 | + (−1.72 − 1.41i)5-s − 0.510i·7-s − 4.78i·11-s − 0.760i·13-s + 4.91·17-s + 5.44i·19-s + 4.00i·23-s + (0.973 + 4.90i)25-s + (5.28 + 1.04i)29-s + 2.99i·31-s + (−0.724 + 0.881i)35-s + 4.18·37-s − 0.105i·41-s − 5.29·43-s + 7.39·47-s + ⋯ |
| L(s) = 1 | + (−0.772 − 0.634i)5-s − 0.192i·7-s − 1.44i·11-s − 0.210i·13-s + 1.19·17-s + 1.24i·19-s + 0.834i·23-s + (0.194 + 0.980i)25-s + (0.980 + 0.194i)29-s + 0.538i·31-s + (−0.122 + 0.149i)35-s + 0.687·37-s − 0.0164i·41-s − 0.808·43-s + 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.667986174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.667986174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.72 + 1.41i)T \) |
| 29 | \( 1 + (-5.28 - 1.04i)T \) |
| good | 7 | \( 1 + 0.510iT - 7T^{2} \) |
| 11 | \( 1 + 4.78iT - 11T^{2} \) |
| 13 | \( 1 + 0.760iT - 13T^{2} \) |
| 17 | \( 1 - 4.91T + 17T^{2} \) |
| 19 | \( 1 - 5.44iT - 19T^{2} \) |
| 23 | \( 1 - 4.00iT - 23T^{2} \) |
| 31 | \( 1 - 2.99iT - 31T^{2} \) |
| 37 | \( 1 - 4.18T + 37T^{2} \) |
| 41 | \( 1 + 0.105iT - 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 - 0.287iT - 53T^{2} \) |
| 59 | \( 1 + 4.33T + 59T^{2} \) |
| 61 | \( 1 + 8.10iT - 61T^{2} \) |
| 67 | \( 1 - 9.29iT - 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 1.25T + 73T^{2} \) |
| 79 | \( 1 - 3.55iT - 79T^{2} \) |
| 83 | \( 1 + 14.9iT - 83T^{2} \) |
| 89 | \( 1 - 5.54iT - 89T^{2} \) |
| 97 | \( 1 + 9.93T + 97T^{2} \) |
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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
−8.111628849932777469954351011229, −7.63754952370160354938471090520, −6.70148422197231688904453624494, −5.69471669453586742261350529202, −5.39672078604891594123056406900, −4.28758675317331685269318941890, −3.55520750840486044774059656671, −3.02944779670441037219630936017, −1.43983953503553695409704082536, −0.66512246049061704109467901817,
0.807429933108417654414449598305, 2.26659029281315447122593525201, 2.85053150694533390624152669560, 3.92876663135952775709081873566, 4.54175702203353544542980129528, 5.27773719015834973261018415892, 6.40127177841212398391096668345, 6.89105780111517101330584341375, 7.59514141459287667677755841469, 8.107082955582424508222340623216
