| L(s) = 1 | − 43.0·2-s − 81·3-s + 1.34e3·4-s + 1.40e3·5-s + 3.48e3·6-s + 9.39e3·7-s − 3.56e4·8-s + 6.56e3·9-s − 6.05e4·10-s − 9.13e4·11-s − 1.08e5·12-s − 9.59e4·13-s − 4.04e5·14-s − 1.13e5·15-s + 8.49e5·16-s − 2.11e5·17-s − 2.82e5·18-s + 5.41e5·19-s + 1.88e6·20-s − 7.60e5·21-s + 3.93e6·22-s − 1.61e6·23-s + 2.89e6·24-s + 2.65e4·25-s + 4.13e6·26-s − 5.31e5·27-s + 1.25e7·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 0.577·3-s + 2.61·4-s + 1.00·5-s + 1.09·6-s + 1.47·7-s − 3.08·8-s + 0.333·9-s − 1.91·10-s − 1.88·11-s − 1.51·12-s − 0.932·13-s − 2.81·14-s − 0.581·15-s + 3.24·16-s − 0.614·17-s − 0.634·18-s + 0.952·19-s + 2.63·20-s − 0.853·21-s + 3.57·22-s − 1.20·23-s + 1.77·24-s + 0.0135·25-s + 1.77·26-s − 0.192·27-s + 3.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6666952683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6666952683\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 + 43.0T + 512T^{2} \) |
| 5 | \( 1 - 1.40e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.39e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 9.13e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.59e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.11e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.41e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.61e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.91e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.46e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.37e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.08e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.24e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.51e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.72e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.44e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.39e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.81e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.89e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.34e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.48e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.28e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.03e8T + 7.60e17T^{2} \) |
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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
−10.71989288674378324212732458540, −10.04557401129377681305569880589, −9.149222429996878661416479432928, −7.81893378080837017682121012843, −7.51154298216042148774770211275, −5.92582556129456486780368487542, −5.10521105673947128142684200674, −2.36863404184670826071344498495, −1.86346061290841794659417460557, −0.53497635808471075067805182492,
0.53497635808471075067805182492, 1.86346061290841794659417460557, 2.36863404184670826071344498495, 5.10521105673947128142684200674, 5.92582556129456486780368487542, 7.51154298216042148774770211275, 7.81893378080837017682121012843, 9.149222429996878661416479432928, 10.04557401129377681305569880589, 10.71989288674378324212732458540
