| L(s) = 1 | − 32·2-s − 745.·3-s + 1.02e3·4-s + 2.38e4·6-s + 7.14e4·7-s − 3.27e4·8-s + 3.78e5·9-s − 3.45e5·11-s − 7.63e5·12-s − 1.50e6·13-s − 2.28e6·14-s + 1.04e6·16-s + 5.39e6·17-s − 1.21e7·18-s − 1.11e7·19-s − 5.33e7·21-s + 1.10e7·22-s + 5.27e6·23-s + 2.44e7·24-s + 4.83e7·26-s − 1.50e8·27-s + 7.32e7·28-s − 1.86e7·29-s + 7.10e7·31-s − 3.35e7·32-s + 2.57e8·33-s − 1.72e8·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.77·3-s + 0.5·4-s + 1.25·6-s + 1.60·7-s − 0.353·8-s + 2.13·9-s − 0.647·11-s − 0.885·12-s − 1.12·13-s − 1.13·14-s + 0.250·16-s + 0.921·17-s − 1.51·18-s − 1.03·19-s − 2.84·21-s + 0.457·22-s + 0.170·23-s + 0.626·24-s + 0.797·26-s − 2.01·27-s + 0.803·28-s − 0.168·29-s + 0.445·31-s − 0.176·32-s + 1.14·33-s − 0.651·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.6915467825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6915467825\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 745.T + 1.77e5T^{2} \) |
| 7 | \( 1 - 7.14e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 3.45e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.50e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.39e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.11e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.27e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.86e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.10e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.23e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 9.11e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.16e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.81e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.05e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.89e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.07e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.70e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 3.45e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.21e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 7.02e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.55e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 9.29e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.71e10T + 7.15e21T^{2} \) |
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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
−12.58697257273047064366454076961, −11.71507379396945688528180149852, −10.86997165167432751045653969930, −10.04212710327705110382342996381, −8.130398260447055630456408112537, −7.04449819738898181160801587065, −5.53788334742394546200910728897, −4.70465236742836023599543065623, −1.88529743289668715644782492150, −0.60743979274823063758381233826,
0.60743979274823063758381233826, 1.88529743289668715644782492150, 4.70465236742836023599543065623, 5.53788334742394546200910728897, 7.04449819738898181160801587065, 8.130398260447055630456408112537, 10.04212710327705110382342996381, 10.86997165167432751045653969930, 11.71507379396945688528180149852, 12.58697257273047064366454076961
