| L(s) = 1 | + 0.670·3-s − 7·7-s − 8.54·9-s − 11·11-s + 24.5·13-s − 20.4·17-s − 4.69·21-s + 25·25-s − 11.7·27-s + 51.0·31-s − 7.37·33-s + 52.6·37-s + 16.4·39-s − 38.9·41-s + 52.6·43-s − 92.0·47-s + 49·49-s − 13.7·51-s + 52.6·53-s + 114.·59-s + 59.1·61-s + 59.8·63-s + 106.·73-s + 16.7·75-s + 77·77-s − 157.·79-s + 69.0·81-s + ⋯ |
| L(s) = 1 | + 0.223·3-s − 7-s − 0.949·9-s − 11-s + 1.88·13-s − 1.20·17-s − 0.223·21-s + 25-s − 0.436·27-s + 1.64·31-s − 0.223·33-s + 1.42·37-s + 0.421·39-s − 0.950·41-s + 1.22·43-s − 1.95·47-s + 0.999·49-s − 0.269·51-s + 0.993·53-s + 1.94·59-s + 0.968·61-s + 0.949·63-s + 1.45·73-s + 0.223·75-s + 77-s − 1.99·79-s + 0.852·81-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.508684313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.508684313\) |
| \(L(2)\) |
|
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 | \( 1 - 0.670T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 13 | \( 1 - 24.5T + 169T^{2} \) |
| 17 | \( 1 + 20.4T + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 51.0T + 961T^{2} \) |
| 37 | \( 1 - 52.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 52.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 92.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 52.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 114.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 59.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 106.T + 5.32e3T^{2} \) |
| 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
−9.460962392573941262433974208456, −8.481519520896572423376632930486, −8.308479250033955031556543510524, −6.84937906418343602764108773942, −6.25537441630081195133523786954, −5.44392088276792494440854671487, −4.21597043107844160179694019987, −3.18852503001976803303757302774, −2.47013344894499896326384720482, −0.69987523740431655165079512414,
0.69987523740431655165079512414, 2.47013344894499896326384720482, 3.18852503001976803303757302774, 4.21597043107844160179694019987, 5.44392088276792494440854671487, 6.25537441630081195133523786954, 6.84937906418343602764108773942, 8.308479250033955031556543510524, 8.481519520896572423376632930486, 9.460962392573941262433974208456
