| L(s) = 1 | − 38.7·2-s + 992.·4-s − 625·5-s + 1.22e4·7-s − 1.86e4·8-s + 2.42e4·10-s + 6.58e4·11-s − 1.12e5·13-s − 4.75e5·14-s + 2.14e5·16-s − 9.66e4·17-s − 1.81e5·19-s − 6.20e5·20-s − 2.55e6·22-s − 2.44e5·23-s + 3.90e5·25-s + 4.35e6·26-s + 1.21e7·28-s + 5.26e6·29-s + 1.83e6·31-s + 1.21e6·32-s + 3.74e6·34-s − 7.67e6·35-s + 6.36e6·37-s + 7.03e6·38-s + 1.16e7·40-s − 1.57e6·41-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.93·4-s − 0.447·5-s + 1.93·7-s − 1.60·8-s + 0.766·10-s + 1.35·11-s − 1.09·13-s − 3.31·14-s + 0.818·16-s − 0.280·17-s − 0.319·19-s − 0.866·20-s − 2.32·22-s − 0.181·23-s + 0.200·25-s + 1.86·26-s + 3.74·28-s + 1.38·29-s + 0.356·31-s + 0.205·32-s + 0.481·34-s − 0.863·35-s + 0.558·37-s + 0.547·38-s + 0.719·40-s − 0.0868·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9851601965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9851601965\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| good | 2 | \( 1 + 38.7T + 512T^{2} \) |
| 7 | \( 1 - 1.22e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.58e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.12e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 9.66e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.81e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.44e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.26e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.83e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.36e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.57e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.99e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.00e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.57e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.19e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.92e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.20e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.99e5T + 4.58e16T^{2} \) |
| 73 | \( 1 + 8.91e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.31e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.69e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.09e6T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.72e8T + 7.60e17T^{2} \) |
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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
−14.32602752364315288338021734041, −11.88946166246651796551249961640, −11.37204484292808353647540385221, −10.10646935639094541940108849076, −8.748233182407995189558827935049, −7.990582164272530402805973693704, −6.85214808409754329638845821226, −4.60354445673882847155388515795, −2.05330947350206793785877014736, −0.896226279663258046420960005613,
0.896226279663258046420960005613, 2.05330947350206793785877014736, 4.60354445673882847155388515795, 6.85214808409754329638845821226, 7.990582164272530402805973693704, 8.748233182407995189558827935049, 10.10646935639094541940108849076, 11.37204484292808353647540385221, 11.88946166246651796551249961640, 14.32602752364315288338021734041
