L(s) = 1 | + 26.7·2-s − 81·3-s + 203.·4-s − 302.·5-s − 2.16e3·6-s − 5.73e3·7-s − 8.24e3·8-s + 6.56e3·9-s − 8.08e3·10-s + 8.93e3·11-s − 1.65e4·12-s − 3.41e4·13-s − 1.53e5·14-s + 2.44e4·15-s − 3.24e5·16-s + 2.51e4·17-s + 1.75e5·18-s + 4.11e4·19-s − 6.15e4·20-s + 4.64e5·21-s + 2.38e5·22-s − 1.88e6·23-s + 6.68e5·24-s − 1.86e6·25-s − 9.13e5·26-s − 5.31e5·27-s − 1.16e6·28-s + ⋯ |
L(s) = 1 | + 1.18·2-s − 0.577·3-s + 0.397·4-s − 0.216·5-s − 0.682·6-s − 0.903·7-s − 0.711·8-s + 0.333·9-s − 0.255·10-s + 0.183·11-s − 0.229·12-s − 0.331·13-s − 1.06·14-s + 0.124·15-s − 1.23·16-s + 0.0730·17-s + 0.394·18-s + 0.0725·19-s − 0.0860·20-s + 0.521·21-s + 0.217·22-s − 1.40·23-s + 0.411·24-s − 0.953·25-s − 0.392·26-s − 0.192·27-s − 0.359·28-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
\(L(5)\) |
\(\approx\) |
\(1.714085912\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.714085912\) |
\(L(\frac{11}{2})\) |
|
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 | \( 1 - 26.7T + 512T^{2} \) |
| 5 | \( 1 + 302.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.73e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.93e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.41e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.51e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.11e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.88e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.48e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.37e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.10e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.38e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.14e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.63e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.61e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.33e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.72e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.07e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.60e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.14e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.37e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.74e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.52e8T + 7.60e17T^{2} \) |
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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
−11.39980943748854060625903813376, −10.09617017171219885096396390410, −9.201529224680573656524110258617, −7.66663970930937014213263058965, −6.36080057201743636675315381612, −5.78864377996456523627927101837, −4.50855610302522454414691775219, −3.70352854429973052367800605046, −2.45380317004302090345267562769, −0.53306618664080250509451749933,
0.53306618664080250509451749933, 2.45380317004302090345267562769, 3.70352854429973052367800605046, 4.50855610302522454414691775219, 5.78864377996456523627927101837, 6.36080057201743636675315381612, 7.66663970930937014213263058965, 9.201529224680573656524110258617, 10.09617017171219885096396390410, 11.39980943748854060625903813376