| L(s) = 1 | + 3.56·5-s + 1.56·7-s + 3.12·11-s − 13-s + 6.68·17-s + 3.12·19-s + 8·23-s + 7.68·25-s − 2·29-s + 4·31-s + 5.56·35-s + 2.68·37-s − 5.12·41-s − 9.56·43-s + 12.6·47-s − 4.56·49-s − 5.12·53-s + 11.1·55-s − 3.12·59-s − 2.87·61-s − 3.56·65-s − 3.12·67-s − 4.68·71-s − 6·73-s + 4.87·77-s + 8·79-s − 14.2·83-s + ⋯ |
| L(s) = 1 | + 1.59·5-s + 0.590·7-s + 0.941·11-s − 0.277·13-s + 1.62·17-s + 0.716·19-s + 1.66·23-s + 1.53·25-s − 0.371·29-s + 0.718·31-s + 0.940·35-s + 0.441·37-s − 0.800·41-s − 1.45·43-s + 1.85·47-s − 0.651·49-s − 0.703·53-s + 1.49·55-s − 0.406·59-s − 0.368·61-s − 0.441·65-s − 0.381·67-s − 0.555·71-s − 0.702·73-s + 0.555·77-s + 0.900·79-s − 1.56·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.881728039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.881728039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 17 | \( 1 - 6.68T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2.68T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 + 9.56T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 + 2.87T + 61T^{2} \) |
| 67 | \( 1 + 3.12T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−7.86033981569702071140177598541, −7.09902422427917036396693827267, −6.48077330120610323932830288298, −5.66133421993093466554656677671, −5.26515144683441848580008645932, −4.52009361374938264981020784821, −3.34199245170124433986065719722, −2.70822429984827463052616679382, −1.52550787057429526120367186833, −1.19231868493006227053018855085,
1.19231868493006227053018855085, 1.52550787057429526120367186833, 2.70822429984827463052616679382, 3.34199245170124433986065719722, 4.52009361374938264981020784821, 5.26515144683441848580008645932, 5.66133421993093466554656677671, 6.48077330120610323932830288298, 7.09902422427917036396693827267, 7.86033981569702071140177598541
