| L(s) = 1 | − 8.77·2-s − 27.1·3-s + 45.0·4-s + 238.·6-s − 203.·7-s − 114.·8-s + 494.·9-s − 494.·11-s − 1.22e3·12-s + 169·13-s + 1.78e3·14-s − 437.·16-s − 1.59e3·17-s − 4.33e3·18-s − 1.10e3·19-s + 5.53e3·21-s + 4.34e3·22-s − 885.·23-s + 3.10e3·24-s − 1.48e3·26-s − 6.82e3·27-s − 9.18e3·28-s + 5.25e3·29-s − 3.15e3·31-s + 7.49e3·32-s + 1.34e4·33-s + 1.40e4·34-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 1.74·3-s + 1.40·4-s + 2.70·6-s − 1.57·7-s − 0.631·8-s + 2.03·9-s − 1.23·11-s − 2.45·12-s + 0.277·13-s + 2.44·14-s − 0.426·16-s − 1.33·17-s − 3.15·18-s − 0.701·19-s + 2.73·21-s + 1.91·22-s − 0.349·23-s + 1.10·24-s − 0.430·26-s − 1.80·27-s − 2.21·28-s + 1.16·29-s − 0.588·31-s + 1.29·32-s + 2.14·33-s + 2.07·34-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 8.77T + 32T^{2} \) |
| 3 | \( 1 + 27.1T + 243T^{2} \) |
| 7 | \( 1 + 203.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 494.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.59e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 885.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.49e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.24e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.31e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.87e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.43e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.45e4T + 8.58e9T^{2} \) |
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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
−10.45290890859236867912060551232, −9.625017700817392240163855732134, −8.588698801411153970675401606414, −7.25407119151884432831654332393, −6.56641050646921648627249301891, −5.79203154459545443041246789685, −4.37743185436214285489145836754, −2.40373868454214745071876353827, −0.68529383039597343362193746658, 0,
0.68529383039597343362193746658, 2.40373868454214745071876353827, 4.37743185436214285489145836754, 5.79203154459545443041246789685, 6.56641050646921648627249301891, 7.25407119151884432831654332393, 8.588698801411153970675401606414, 9.625017700817392240163855732134, 10.45290890859236867912060551232
