| L(s) = 1 | − 5.21·2-s + 243·3-s − 2.02e3·4-s − 8.55e3·5-s − 1.26e3·6-s − 1.65e4·7-s + 2.12e4·8-s + 5.90e4·9-s + 4.45e4·10-s − 8.92e4·11-s − 4.91e5·12-s − 1.14e5·13-s + 8.61e4·14-s − 2.07e6·15-s + 4.02e6·16-s − 2.27e6·17-s − 3.07e5·18-s − 1.31e7·19-s + 1.72e7·20-s − 4.01e6·21-s + 4.65e5·22-s − 2.62e7·23-s + 5.15e6·24-s + 2.42e7·25-s + 5.98e5·26-s + 1.43e7·27-s + 3.33e7·28-s + ⋯ |
| L(s) = 1 | − 0.115·2-s + 0.577·3-s − 0.986·4-s − 1.22·5-s − 0.0664·6-s − 0.371·7-s + 0.228·8-s + 0.333·9-s + 0.140·10-s − 0.167·11-s − 0.569·12-s − 0.0857·13-s + 0.0427·14-s − 0.706·15-s + 0.960·16-s − 0.388·17-s − 0.0383·18-s − 1.22·19-s + 1.20·20-s − 0.214·21-s + 0.0192·22-s − 0.851·23-s + 0.132·24-s + 0.497·25-s + 0.00987·26-s + 0.192·27-s + 0.366·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.2283791978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2283791978\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 + 5.21T + 2.04e3T^{2} \) |
| 5 | \( 1 + 8.55e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 1.65e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 8.92e4T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.14e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 2.27e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.31e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.62e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.02e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.81e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 9.42e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.20e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 6.09e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.59e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.94e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 4.54e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 9.68e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.70e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 8.07e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.36e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.87e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.72e9T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.22e11T + 7.15e21T^{2} \) |
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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
−10.51790105292250625577626427648, −9.506084690152816564753051926165, −8.527048687078293915510730050977, −7.949597092538637886177135320704, −6.81090064117247315343238183129, −5.20486486276053989094035580127, −4.01686775132586466453326184273, −3.53348936756059100472626588248, −1.90072246352794431046007105346, −0.20783137167557155514127006537,
0.20783137167557155514127006537, 1.90072246352794431046007105346, 3.53348936756059100472626588248, 4.01686775132586466453326184273, 5.20486486276053989094035580127, 6.81090064117247315343238183129, 7.949597092538637886177135320704, 8.527048687078293915510730050977, 9.506084690152816564753051926165, 10.51790105292250625577626427648
