| L(s) = 1 | + 8·2-s − 53.5·3-s + 64·4-s − 314.·5-s − 428.·6-s − 1.16e3·7-s + 512·8-s + 675.·9-s − 2.51e3·10-s + 3.43e3·11-s − 3.42e3·12-s + 1.09e4·13-s − 9.31e3·14-s + 1.68e4·15-s + 4.09e3·16-s − 6.91e3·17-s + 5.40e3·18-s + 3.52e3·19-s − 2.01e4·20-s + 6.22e4·21-s + 2.74e4·22-s − 5.43e4·23-s − 2.73e4·24-s + 2.10e4·25-s + 8.73e4·26-s + 8.08e4·27-s − 7.45e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.14·3-s + 0.5·4-s − 1.12·5-s − 0.809·6-s − 1.28·7-s + 0.353·8-s + 0.309·9-s − 0.796·10-s + 0.778·11-s − 0.572·12-s + 1.37·13-s − 0.907·14-s + 1.28·15-s + 0.250·16-s − 0.341·17-s + 0.218·18-s + 0.117·19-s − 0.563·20-s + 1.46·21-s + 0.550·22-s − 0.931·23-s − 0.404·24-s + 0.269·25-s + 0.974·26-s + 0.790·27-s − 0.641·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.235532185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.235532185\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 37 | \( 1 - 5.06e4T \) |
| good | 3 | \( 1 + 53.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 314.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.16e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.43e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.09e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 6.91e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.52e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.43e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.58e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.93e5T + 2.75e10T^{2} \) |
| 41 | \( 1 - 4.79e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.00e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.32e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.81e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.37e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.85e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.98e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.66e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.57e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.09e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.25e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.06e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.53e7T + 8.07e13T^{2} \) |
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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
−12.89913779677346345309491485500, −11.88320978913946296517962023016, −11.39717956886834694264403059226, −10.12233235668272731383308318420, −8.372338857126345129548462308586, −6.63877062187775463206450503666, −6.10526396147929208030885755746, −4.39751728168748794323009539412, −3.32615097875466929924972285720, −0.69880199121176732574555336500,
0.69880199121176732574555336500, 3.32615097875466929924972285720, 4.39751728168748794323009539412, 6.10526396147929208030885755746, 6.63877062187775463206450503666, 8.372338857126345129548462308586, 10.12233235668272731383308318420, 11.39717956886834694264403059226, 11.88320978913946296517962023016, 12.89913779677346345309491485500
