| L(s) = 1 | − 122.·7-s + 100·11-s − 734.·13-s − 979.·17-s − 2.24e3·19-s − 3.41e3·23-s + 7.85e3·29-s − 2.14e3·31-s + 1.04e4·37-s + 7.41e3·41-s − 1.77e4·43-s + 9.43e3·47-s − 1.80e3·49-s − 2.42e4·53-s − 2.59e4·59-s − 3.05e3·61-s + 5.87e4·67-s − 3.76e4·71-s − 2.40e4·73-s − 1.22e4·77-s + 7.97e4·79-s + 1.62e4·83-s + 826·89-s + 9.00e4·91-s − 3.75e4·97-s + 1.43e5·101-s + 1.11e5·103-s + ⋯ |
| L(s) = 1 | − 0.944·7-s + 0.249·11-s − 1.20·13-s − 0.822·17-s − 1.42·19-s − 1.34·23-s + 1.73·29-s − 0.400·31-s + 1.24·37-s + 0.688·41-s − 1.46·43-s + 0.622·47-s − 0.107·49-s − 1.18·53-s − 0.971·59-s − 0.105·61-s + 1.59·67-s − 0.885·71-s − 0.527·73-s − 0.235·77-s + 1.43·79-s + 0.259·83-s + 0.0110·89-s + 1.13·91-s − 0.405·97-s + 1.40·101-s + 1.03·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8535634160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8535634160\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 122.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 100T + 1.61e5T^{2} \) |
| 13 | \( 1 + 734.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 979.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.04e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.77e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.43e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.42e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.05e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 826T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.75e4T + 8.58e9T^{2} \) |
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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
−9.492762286111604907630712478312, −8.560755064139141574280447513051, −7.69574449252001483268063421808, −6.56742657931583114749561474821, −6.23241822505047085008859907743, −4.82734919442811781739935113052, −4.07919677718428348735016542640, −2.85788835094588319723092325705, −2.00635172279950824015359818170, −0.38744294117697604468310621140,
0.38744294117697604468310621140, 2.00635172279950824015359818170, 2.85788835094588319723092325705, 4.07919677718428348735016542640, 4.82734919442811781739935113052, 6.23241822505047085008859907743, 6.56742657931583114749561474821, 7.69574449252001483268063421808, 8.560755064139141574280447513051, 9.492762286111604907630712478312
