| L(s) = 1 | + 8.76·2-s + 44.8·4-s + 82.0·5-s + 213.·7-s + 112.·8-s + 719.·10-s + 252.·11-s − 310.·13-s + 1.87e3·14-s − 449.·16-s − 2.19e3·17-s + 646.·19-s + 3.67e3·20-s + 2.21e3·22-s − 529·23-s + 3.60e3·25-s − 2.72e3·26-s + 9.56e3·28-s + 696.·29-s + 2.86e3·31-s − 7.53e3·32-s − 1.92e4·34-s + 1.75e4·35-s + 6.79e3·37-s + 5.66e3·38-s + 9.21e3·40-s − 2.97e3·41-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 1.40·4-s + 1.46·5-s + 1.64·7-s + 0.620·8-s + 2.27·10-s + 0.629·11-s − 0.509·13-s + 2.55·14-s − 0.438·16-s − 1.84·17-s + 0.410·19-s + 2.05·20-s + 0.975·22-s − 0.208·23-s + 1.15·25-s − 0.789·26-s + 2.30·28-s + 0.153·29-s + 0.534·31-s − 1.30·32-s − 2.86·34-s + 2.41·35-s + 0.816·37-s + 0.636·38-s + 0.911·40-s − 0.276·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.928834764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.928834764\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 - 8.76T + 32T^{2} \) |
| 5 | \( 1 - 82.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 213.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 252.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 310.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.19e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 646.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 696.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.97e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.13e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.51e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.64e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.31e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.51e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.24e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.32e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.88e3T + 8.58e9T^{2} \) |
| 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
−11.61877523970097495857687838118, −10.99281142697654189086681700846, −9.633785980918590069553508485115, −8.536520418860676436078010096442, −6.95105880166181100550869348028, −6.00404967005910137880146368230, −5.02640975741177559219182386546, −4.34555140736302925777052593161, −2.50926948154661022833512238430, −1.66543597872120842819948881021,
1.66543597872120842819948881021, 2.50926948154661022833512238430, 4.34555140736302925777052593161, 5.02640975741177559219182386546, 6.00404967005910137880146368230, 6.95105880166181100550869348028, 8.536520418860676436078010096442, 9.633785980918590069553508485115, 10.99281142697654189086681700846, 11.61877523970097495857687838118
