| L(s) = 1 | + 40.0·2-s + 243.·3-s + 1.09e3·4-s + 1.21e3·5-s + 9.75e3·6-s − 1.20e4·7-s + 2.33e4·8-s + 3.95e4·9-s + 4.85e4·10-s − 5.73e3·11-s + 2.66e5·12-s − 1.19e5·13-s − 4.81e5·14-s + 2.94e5·15-s + 3.76e5·16-s + 1.88e4·17-s + 1.58e6·18-s − 8.01e5·19-s + 1.32e6·20-s − 2.92e6·21-s − 2.29e5·22-s + 1.60e6·23-s + 5.68e6·24-s − 4.84e5·25-s − 4.80e6·26-s + 4.82e6·27-s − 1.31e7·28-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 1.73·3-s + 2.13·4-s + 0.867·5-s + 3.07·6-s − 1.89·7-s + 2.01·8-s + 2.00·9-s + 1.53·10-s − 0.118·11-s + 3.70·12-s − 1.16·13-s − 3.34·14-s + 1.50·15-s + 1.43·16-s + 0.0546·17-s + 3.55·18-s − 1.41·19-s + 1.85·20-s − 3.27·21-s − 0.209·22-s + 1.19·23-s + 3.49·24-s − 0.248·25-s − 2.06·26-s + 1.74·27-s − 4.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(8.676291870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.676291870\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 - 40.0T + 512T^{2} \) |
| 3 | \( 1 - 243.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.21e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.20e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.73e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.19e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.88e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.01e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.60e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.51e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.59e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.09e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.47e7T + 3.27e14T^{2} \) |
| 47 | \( 1 - 3.38e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.59e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.13e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.16e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.90e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.00e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.72e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.01e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.19e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.19e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.48e8T + 7.60e17T^{2} \) |
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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
−13.78715254906469780048466287839, −13.12025406513934572456882266865, −12.54827645209759173978253835296, −10.17649771624481170614877267597, −9.214150948793644009945046406006, −7.20197314295011632735798720232, −6.11557884716611764483172497056, −4.24750549642198569585161166911, −2.93505626480794264876763720517, −2.38905313246973896845043014507,
2.38905313246973896845043014507, 2.93505626480794264876763720517, 4.24750549642198569585161166911, 6.11557884716611764483172497056, 7.20197314295011632735798720232, 9.214150948793644009945046406006, 10.17649771624481170614877267597, 12.54827645209759173978253835296, 13.12025406513934572456882266865, 13.78715254906469780048466287839
