| L(s) = 1 | + 9.61·2-s − 9·3-s + 60.3·4-s + 25·5-s − 86.5·6-s − 217.·7-s + 272.·8-s + 81·9-s + 240.·10-s − 199.·11-s − 543.·12-s + 599.·13-s − 2.09e3·14-s − 225·15-s + 690.·16-s + 209.·17-s + 778.·18-s + 2.83e3·19-s + 1.50e3·20-s + 1.96e3·21-s − 1.91e3·22-s − 2.09e3·23-s − 2.45e3·24-s + 625·25-s + 5.76e3·26-s − 729·27-s − 1.31e4·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s − 0.577·3-s + 1.88·4-s + 0.447·5-s − 0.981·6-s − 1.67·7-s + 1.50·8-s + 0.333·9-s + 0.759·10-s − 0.497·11-s − 1.08·12-s + 0.984·13-s − 2.85·14-s − 0.258·15-s + 0.674·16-s + 0.175·17-s + 0.566·18-s + 1.80·19-s + 0.843·20-s + 0.969·21-s − 0.845·22-s − 0.825·23-s − 0.870·24-s + 0.200·25-s + 1.67·26-s − 0.192·27-s − 3.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.438245893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.438245893\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 - 25T \) |
| good | 2 | \( 1 - 9.61T + 32T^{2} \) |
| 7 | \( 1 + 217.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 199.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 599.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 209.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.83e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 326.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.91e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.31e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.57e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.21e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.74e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.49e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.18e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.13e5T + 8.58e9T^{2} \) |
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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
−18.34102449283070245774059162180, −16.30909172196001982194546056336, −15.68714642783563040073595455386, −13.77763231335285740308837784028, −13.01608575580521846388314692487, −11.78376501848841984784445697619, −10.01870085045221936714909375777, −6.64807196065025648412328131646, −5.53636252312663815802493812055, −3.35445353055115695514641053987,
3.35445353055115695514641053987, 5.53636252312663815802493812055, 6.64807196065025648412328131646, 10.01870085045221936714909375777, 11.78376501848841984784445697619, 13.01608575580521846388314692487, 13.77763231335285740308837784028, 15.68714642783563040073595455386, 16.30909172196001982194546056336, 18.34102449283070245774059162180
