L(s) = 1 | + 5.96·3-s − 7·7-s + 26.5·9-s − 11·11-s + 8.64·13-s + 27.1·17-s − 41.7·21-s + 25·25-s + 104.·27-s + 35.1·31-s − 65.5·33-s − 52.6·37-s + 51.5·39-s + 72.1·41-s − 52.6·43-s + 19.0·47-s + 49·49-s + 161.·51-s − 52.6·53-s − 28.3·59-s + 106.·61-s − 185.·63-s − 99.8·73-s + 149.·75-s + 77·77-s + 157.·79-s + 384.·81-s + ⋯ |
L(s) = 1 | + 1.98·3-s − 7-s + 2.94·9-s − 11-s + 0.665·13-s + 1.59·17-s − 1.98·21-s + 25-s + 3.87·27-s + 1.13·31-s − 1.98·33-s − 1.42·37-s + 1.32·39-s + 1.75·41-s − 1.22·43-s + 0.405·47-s + 0.999·49-s + 3.17·51-s − 0.993·53-s − 0.479·59-s + 1.74·61-s − 2.94·63-s − 1.36·73-s + 1.98·75-s + 77-s + 1.99·79-s + 4.75·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(4.031085153\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.031085153\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
good | 3 | \( 1 - 5.96T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 13 | \( 1 - 8.64T + 169T^{2} \) |
| 17 | \( 1 - 27.1T + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 35.1T + 961T^{2} \) |
| 37 | \( 1 + 52.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 72.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 52.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 19.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 52.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 28.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 106.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 99.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 157.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.474647274123102285653628143344, −8.643109966119129767690224470202, −8.038007389043707786323139492700, −7.32645246329711362584354092032, −6.45633757927332486480338454369, −5.14120241212597921199381859200, −3.89955662496109751248162859781, −3.17385597132120420944542159349, −2.58368150229558019645112819919, −1.18068152680698404969880027701,
1.18068152680698404969880027701, 2.58368150229558019645112819919, 3.17385597132120420944542159349, 3.89955662496109751248162859781, 5.14120241212597921199381859200, 6.45633757927332486480338454369, 7.32645246329711362584354092032, 8.038007389043707786323139492700, 8.643109966119129767690224470202, 9.474647274123102285653628143344