| L(s) = 1 | − 2.89·2-s + 3.45·3-s − 23.6·4-s − 9.99·6-s + 148.·7-s + 160.·8-s − 231.·9-s − 712.·11-s − 81.7·12-s − 169·13-s − 428.·14-s + 290.·16-s + 1.13e3·17-s + 668.·18-s + 1.40e3·19-s + 512.·21-s + 2.06e3·22-s − 897.·23-s + 556.·24-s + 488.·26-s − 1.63e3·27-s − 3.50e3·28-s + 3.23e3·29-s − 7.97e3·31-s − 5.99e3·32-s − 2.46e3·33-s − 3.27e3·34-s + ⋯ |
| L(s) = 1 | − 0.511·2-s + 0.221·3-s − 0.738·4-s − 0.113·6-s + 1.14·7-s + 0.888·8-s − 0.950·9-s − 1.77·11-s − 0.163·12-s − 0.277·13-s − 0.584·14-s + 0.284·16-s + 0.949·17-s + 0.486·18-s + 0.889·19-s + 0.253·21-s + 0.907·22-s − 0.353·23-s + 0.197·24-s + 0.141·26-s − 0.432·27-s − 0.844·28-s + 0.714·29-s − 1.49·31-s − 1.03·32-s − 0.393·33-s − 0.485·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.083147343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083147343\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 + 2.89T + 32T^{2} \) |
| 3 | \( 1 - 3.45T + 243T^{2} \) |
| 7 | \( 1 - 148.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 712.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.13e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.40e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 897.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.55e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.60e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.51e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.85e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.98e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.38e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.92e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.47e5T + 8.58e9T^{2} \) |
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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
−10.58838080428026797176773395402, −9.865919483985602712838639697284, −8.689922834709196654821805393069, −8.062874994830055512684357685190, −7.46933406952916196990672282667, −5.32146356531259743334083395933, −5.16862447338970500433484816340, −3.49396614471559063584036307735, −2.11790925238859219760816648974, −0.61147354841971440753823090435,
0.61147354841971440753823090435, 2.11790925238859219760816648974, 3.49396614471559063584036307735, 5.16862447338970500433484816340, 5.32146356531259743334083395933, 7.46933406952916196990672282667, 8.062874994830055512684357685190, 8.689922834709196654821805393069, 9.865919483985602712838639697284, 10.58838080428026797176773395402
