| L(s) = 1 | − 10.4·2-s + 76.6·4-s − 25·5-s − 101.·7-s − 465.·8-s + 260.·10-s − 121·11-s + 463.·13-s + 1.05e3·14-s + 2.39e3·16-s − 1.57e3·17-s − 2.43e3·19-s − 1.91e3·20-s + 1.26e3·22-s − 2.76e3·23-s + 625·25-s − 4.82e3·26-s − 7.75e3·28-s + 5.69e3·29-s − 5.75e3·31-s − 1.00e4·32-s + 1.63e4·34-s + 2.53e3·35-s − 2.75e3·37-s + 2.53e4·38-s + 1.16e4·40-s + 1.92e4·41-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 2.39·4-s − 0.447·5-s − 0.780·7-s − 2.56·8-s + 0.823·10-s − 0.301·11-s + 0.760·13-s + 1.43·14-s + 2.33·16-s − 1.31·17-s − 1.54·19-s − 1.07·20-s + 0.555·22-s − 1.09·23-s + 0.200·25-s − 1.40·26-s − 1.86·28-s + 1.25·29-s − 1.07·31-s − 1.73·32-s + 2.43·34-s + 0.349·35-s − 0.330·37-s + 2.84·38-s + 1.14·40-s + 1.78·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2024748520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2024748520\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 10.4T + 32T^{2} \) |
| 7 | \( 1 + 101.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 463.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.57e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.43e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.76e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.92e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.08e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.31e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.28e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.60e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.06e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.18e5T + 8.58e9T^{2} \) |
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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
−10.12726282111339139217478959277, −9.077740850795476041830534213816, −8.551086102503501885752270308351, −7.72183157013281028462211641309, −6.65632572339368393362114867455, −6.15460926048755182557117989865, −4.19483114125950928740678261470, −2.81415893222212339812095484799, −1.74519204592404206692060871448, −0.28906637521854905288528438193,
0.28906637521854905288528438193, 1.74519204592404206692060871448, 2.81415893222212339812095484799, 4.19483114125950928740678261470, 6.15460926048755182557117989865, 6.65632572339368393362114867455, 7.72183157013281028462211641309, 8.551086102503501885752270308351, 9.077740850795476041830534213816, 10.12726282111339139217478959277
