| L(s) = 1 | + 0.378·7-s − 3.48·13-s − 3.46·19-s − 5·25-s − 10.1·31-s + 2.17·37-s + 10.3·43-s − 6.85·49-s + 14.7·61-s + 16·67-s + 13.8·73-s + 9.41·79-s − 1.32·91-s + 13.8·97-s + 11.6·103-s − 16.1·109-s + ⋯ |
| L(s) = 1 | + 0.143·7-s − 0.966·13-s − 0.794·19-s − 25-s − 1.82·31-s + 0.357·37-s + 1.58·43-s − 0.979·49-s + 1.89·61-s + 1.95·67-s + 1.62·73-s + 1.05·79-s − 0.138·91-s + 1.40·97-s + 1.15·103-s − 1.54·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.491338504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.491338504\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 0.378T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 - 16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 9.41T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77438046370802669396313892625, −7.05219331767243649611381422107, −6.41825802608932836101828822263, −5.56331382709436321382375775721, −5.05121838277036801930042013401, −4.14053414889633171271156859426, −3.58518650484385927523282139172, −2.42505165566482354671829267791, −1.93481572947072796841943348901, −0.56507748145586868443495279768,
0.56507748145586868443495279768, 1.93481572947072796841943348901, 2.42505165566482354671829267791, 3.58518650484385927523282139172, 4.14053414889633171271156859426, 5.05121838277036801930042013401, 5.56331382709436321382375775721, 6.41825802608932836101828822263, 7.05219331767243649611381422107, 7.77438046370802669396313892625
