| L(s) = 1 | + 32·2-s + 243·3-s + 1.02e3·4-s + 3.12e3·5-s + 7.77e3·6-s + 1.68e4·7-s + 3.27e4·8-s + 5.90e4·9-s + 1.00e5·10-s − 8.02e5·11-s + 2.48e5·12-s + 1.29e6·13-s + 5.37e5·14-s + 7.59e5·15-s + 1.04e6·16-s + 4.53e6·17-s + 1.88e6·18-s + 2.73e6·19-s + 3.20e6·20-s + 4.08e6·21-s − 2.56e7·22-s − 2.07e7·23-s + 7.96e6·24-s + 9.76e6·25-s + 4.14e7·26-s + 1.43e7·27-s + 1.72e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.50·11-s + 0.288·12-s + 0.967·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s + 0.773·17-s + 0.235·18-s + 0.253·19-s + 0.223·20-s + 0.218·21-s − 1.06·22-s − 0.671·23-s + 0.204·24-s + 0.199·25-s + 0.684·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(5.718142507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.718142507\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 3 | \( 1 - 243T \) |
| 5 | \( 1 - 3.12e3T \) |
| 7 | \( 1 - 1.68e4T \) |
| good | 11 | \( 1 + 8.02e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.29e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.53e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.73e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.07e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 3.95e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 8.36e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.45e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.42e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.63e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.57e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.33e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.95e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 9.07e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 4.11e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.60e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.40e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.82e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.93e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.47e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 9.83e10T + 7.15e21T^{2} \) |
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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
−10.48969507659095137671922918644, −9.490023763763834824580980737061, −8.176682082635177995267039814354, −7.55986661963145432283582827239, −6.11259066250300813454781854460, −5.31422801123583529060166105933, −4.15540382998205030557410637441, −3.00335825802653935839394822545, −2.14761690749961516454598485638, −0.939201699977615044625180610938,
0.939201699977615044625180610938, 2.14761690749961516454598485638, 3.00335825802653935839394822545, 4.15540382998205030557410637441, 5.31422801123583529060166105933, 6.11259066250300813454781854460, 7.55986661963145432283582827239, 8.176682082635177995267039814354, 9.490023763763834824580980737061, 10.48969507659095137671922918644
