| L(s) = 1 | − 16.4·2-s − 242.·4-s − 702.·5-s − 3.56e3·7-s + 1.23e4·8-s + 1.15e4·10-s + 3.38e4·11-s − 2.28e4·13-s + 5.85e4·14-s − 7.95e4·16-s + 7.19e4·17-s − 1.30e5·19-s + 1.70e5·20-s − 5.56e5·22-s − 7.58e5·23-s − 1.45e6·25-s + 3.74e5·26-s + 8.62e5·28-s − 6.67e6·29-s + 2.64e6·31-s − 5.03e6·32-s − 1.18e6·34-s + 2.50e6·35-s − 8.03e6·37-s + 2.14e6·38-s − 8.70e6·40-s + 2.60e7·41-s + ⋯ |
| L(s) = 1 | − 0.726·2-s − 0.472·4-s − 0.502·5-s − 0.560·7-s + 1.06·8-s + 0.365·10-s + 0.697·11-s − 0.221·13-s + 0.407·14-s − 0.303·16-s + 0.208·17-s − 0.229·19-s + 0.237·20-s − 0.506·22-s − 0.565·23-s − 0.747·25-s + 0.160·26-s + 0.265·28-s − 1.75·29-s + 0.514·31-s − 0.849·32-s − 0.151·34-s + 0.282·35-s − 0.704·37-s + 0.166·38-s − 0.537·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5644209044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5644209044\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 + 16.4T + 512T^{2} \) |
| 5 | \( 1 + 702.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.56e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.38e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.28e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 7.19e4T + 1.18e11T^{2} \) |
| 23 | \( 1 + 7.58e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.67e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.64e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.03e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.60e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.94e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.50e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.48e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.87e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.11e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.98e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 5.48e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.09e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.80e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.46e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.00e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.16e8T + 7.60e17T^{2} \) |
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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
−10.90741412352621182109317185275, −9.807576275318366693555016835012, −9.177035831582935952264393463389, −8.090616142188245997144773704286, −7.23629487453571796886688284794, −5.89895179682259501757060642611, −4.44464785004045418716312807342, −3.52322405152458787028951214881, −1.75853167905882841021963673727, −0.42063026014026212245958206744,
0.42063026014026212245958206744, 1.75853167905882841021963673727, 3.52322405152458787028951214881, 4.44464785004045418716312807342, 5.89895179682259501757060642611, 7.23629487453571796886688284794, 8.090616142188245997144773704286, 9.177035831582935952264393463389, 9.807576275318366693555016835012, 10.90741412352621182109317185275
