| L(s) = 1 | + 2.67·3-s − 1.86·9-s + 11·11-s + 45.2·23-s − 29.0·27-s + 24.5·31-s + 29.3·33-s + 13.1·37-s + 79.5·47-s + 49·49-s − 79.5·53-s + 96.5·59-s − 34.3·67-s + 120.·69-s + 23.4·71-s − 60.7·81-s + 177.·89-s + 65.6·93-s + 2.30·97-s − 20.4·99-s − 79.5·103-s + 35.1·111-s − 221.·113-s + ⋯ |
| L(s) = 1 | + 0.890·3-s − 0.206·9-s + 11-s + 1.96·23-s − 1.07·27-s + 0.793·31-s + 0.890·33-s + 0.356·37-s + 1.69·47-s + 0.999·49-s − 1.50·53-s + 1.63·59-s − 0.512·67-s + 1.75·69-s + 0.329·71-s − 0.750·81-s + 1.99·89-s + 0.706·93-s + 0.0237·97-s − 0.206·99-s − 0.772·103-s + 0.317·111-s − 1.95·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.805627717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.805627717\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 2.67T + 9T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 45.2T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 24.5T + 961T^{2} \) |
| 37 | \( 1 - 13.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 79.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 79.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 96.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 34.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 23.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 177.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2.30T + 9.40e3T^{2} \) |
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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
−9.277564181855092347655457070701, −9.014850342381472937857022574408, −8.133293768379835084547190381568, −7.24087720797753512994958049469, −6.41663406836379611528668241676, −5.35298762866890582872920365700, −4.23990939906990944371510118665, −3.29987879451870537324516319151, −2.41678073056925611142483919343, −1.02799407163904526391667560502,
1.02799407163904526391667560502, 2.41678073056925611142483919343, 3.29987879451870537324516319151, 4.23990939906990944371510118665, 5.35298762866890582872920365700, 6.41663406836379611528668241676, 7.24087720797753512994958049469, 8.133293768379835084547190381568, 9.014850342381472937857022574408, 9.277564181855092347655457070701
