| L(s) = 1 | − 4.78e3·2-s + 1.44e7·4-s − 1.42e8·5-s + 6.49e9·7-s − 2.92e10·8-s + 6.81e11·10-s + 6.56e11·11-s − 1.03e13·13-s − 3.10e13·14-s + 1.81e13·16-s − 2.36e14·17-s + 5.53e14·19-s − 2.06e15·20-s − 3.14e15·22-s − 3.36e15·23-s + 8.39e15·25-s + 4.93e16·26-s + 9.41e16·28-s + 8.75e15·29-s − 1.61e17·31-s + 1.58e17·32-s + 1.12e18·34-s − 9.25e17·35-s + 1.42e18·37-s − 2.64e18·38-s + 4.16e18·40-s + 4.86e18·41-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.72·4-s − 1.30·5-s + 1.24·7-s − 1.20·8-s + 2.15·10-s + 0.694·11-s − 1.59·13-s − 2.05·14-s + 0.258·16-s − 1.67·17-s + 1.08·19-s − 2.25·20-s − 1.14·22-s − 0.736·23-s + 0.703·25-s + 2.63·26-s + 2.14·28-s + 0.133·29-s − 1.14·31-s + 0.775·32-s + 2.76·34-s − 1.62·35-s + 1.31·37-s − 1.79·38-s + 1.57·40-s + 1.38·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.5372300160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5372300160\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 4.78e3T + 8.38e6T^{2} \) |
| 5 | \( 1 + 1.42e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 6.49e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 6.56e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 1.03e13T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.36e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 5.53e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 3.36e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 8.75e15T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.61e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.42e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 4.86e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 3.63e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 1.28e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 1.00e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.29e19T + 5.36e40T^{2} \) |
| 61 | \( 1 - 1.60e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 3.50e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.05e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 9.56e20T + 7.18e42T^{2} \) |
| 79 | \( 1 + 4.09e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.62e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 2.59e21T + 6.85e44T^{2} \) |
| 97 | \( 1 - 2.98e22T + 4.96e45T^{2} \) |
| 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
−15.93731869148210211975640204774, −14.68145745779802467254031678199, −11.84944150167094798669609428740, −11.08960936023016417287266044163, −9.336338677270473833491085338775, −8.016192216748621590815183854125, −7.21904631428505311504623563798, −4.45205268515798882569664567042, −2.11163952357875927157202795494, −0.58020662713304802923910780443,
0.58020662713304802923910780443, 2.11163952357875927157202795494, 4.45205268515798882569664567042, 7.21904631428505311504623563798, 8.016192216748621590815183854125, 9.336338677270473833491085338775, 11.08960936023016417287266044163, 11.84944150167094798669609428740, 14.68145745779802467254031678199, 15.93731869148210211975640204774
