| L(s) = 1 | − 0.743·2-s + 3-s − 1.44·4-s + 0.602·5-s − 0.743·6-s + 4.42·7-s + 2.56·8-s + 9-s − 0.448·10-s − 2.69·11-s − 1.44·12-s − 5.35·13-s − 3.28·14-s + 0.602·15-s + 0.986·16-s − 17-s − 0.743·18-s + 1.87·19-s − 0.871·20-s + 4.42·21-s + 2.00·22-s + 1.99·23-s + 2.56·24-s − 4.63·25-s + 3.98·26-s + 27-s − 6.39·28-s + ⋯ |
| L(s) = 1 | − 0.526·2-s + 0.577·3-s − 0.723·4-s + 0.269·5-s − 0.303·6-s + 1.67·7-s + 0.906·8-s + 0.333·9-s − 0.141·10-s − 0.813·11-s − 0.417·12-s − 1.48·13-s − 0.879·14-s + 0.155·15-s + 0.246·16-s − 0.242·17-s − 0.175·18-s + 0.429·19-s − 0.194·20-s + 0.965·21-s + 0.427·22-s + 0.415·23-s + 0.523·24-s − 0.927·25-s + 0.781·26-s + 0.192·27-s − 1.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 0.743T + 2T^{2} \) |
| 5 | \( 1 - 0.602T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 - 1.99T + 23T^{2} \) |
| 29 | \( 1 - 1.22T + 29T^{2} \) |
| 31 | \( 1 + 7.92T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 2.98T + 43T^{2} \) |
| 47 | \( 1 - 0.285T + 47T^{2} \) |
| 53 | \( 1 + 9.29T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 - 1.12T + 71T^{2} \) |
| 73 | \( 1 + 8.28T + 73T^{2} \) |
| 83 | \( 1 + 3.78T + 83T^{2} \) |
| 89 | \( 1 + 5.73T + 89T^{2} \) |
| 97 | \( 1 - 5.52T + 97T^{2} \) |
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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
−8.133085100910790056838561807589, −7.57170698122070005578166454510, −7.08949352082453853015198865678, −5.45535245455321131379734464912, −5.05160761615605028721522793101, −4.47226973197294824185622321411, −3.37789667948163839176214170946, −2.14867249149034735266788509218, −1.57215714693519747902144870542, 0,
1.57215714693519747902144870542, 2.14867249149034735266788509218, 3.37789667948163839176214170946, 4.47226973197294824185622321411, 5.05160761615605028721522793101, 5.45535245455321131379734464912, 7.08949352082453853015198865678, 7.57170698122070005578166454510, 8.133085100910790056838561807589
