| L(s) = 1 | − 16·2-s − 144.·3-s + 256·4-s − 1.70e3·5-s + 2.31e3·6-s − 8.66e3·7-s − 4.09e3·8-s + 1.24e3·9-s + 2.73e4·10-s + 1.46e4·11-s − 3.70e4·12-s + 6.78e4·13-s + 1.38e5·14-s + 2.46e5·15-s + 6.55e4·16-s + 2.49e5·17-s − 1.99e4·18-s − 2.93e5·19-s − 4.36e5·20-s + 1.25e6·21-s − 2.34e5·22-s + 1.10e6·23-s + 5.92e5·24-s + 9.59e5·25-s − 1.08e6·26-s + 2.66e6·27-s − 2.21e6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.03·3-s + 0.5·4-s − 1.22·5-s + 0.729·6-s − 1.36·7-s − 0.353·8-s + 0.0633·9-s + 0.863·10-s + 0.301·11-s − 0.515·12-s + 0.658·13-s + 0.964·14-s + 1.25·15-s + 0.250·16-s + 0.724·17-s − 0.0448·18-s − 0.516·19-s − 0.610·20-s + 1.40·21-s − 0.213·22-s + 0.821·23-s + 0.364·24-s + 0.491·25-s − 0.465·26-s + 0.965·27-s − 0.681·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.3452052451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3452052451\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 11 | \( 1 - 1.46e4T \) |
| good | 3 | \( 1 + 144.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.70e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.66e3T + 4.03e7T^{2} \) |
| 13 | \( 1 - 6.78e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.49e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.93e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.10e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.23e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.34e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.01e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.56e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.88e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.82e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.22e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.80e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.67e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.51e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.95e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.61e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.63e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.06e8T + 7.60e17T^{2} \) |
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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
−16.33866143523375467600717014945, −15.11972734325486814328519758322, −12.79303822713282876366190642448, −11.68806574079120524385995203303, −10.68515759156517896957359790126, −9.039668237516440359554431463109, −7.30359298897399984863097056950, −5.95245196720830927398293647809, −3.56023544781123300836964739338, −0.51678432464801416817963132739,
0.51678432464801416817963132739, 3.56023544781123300836964739338, 5.95245196720830927398293647809, 7.30359298897399984863097056950, 9.039668237516440359554431463109, 10.68515759156517896957359790126, 11.68806574079120524385995203303, 12.79303822713282876366190642448, 15.11972734325486814328519758322, 16.33866143523375467600717014945
