| L(s) = 1 | − 2·2-s + 5.89·3-s + 4·4-s − 11.7·6-s − 8·8-s + 25.7·9-s + 7.69·11-s + 23.5·12-s + 16·16-s − 30.3·17-s − 51.5·18-s + 2.30·19-s − 15.3·22-s − 47.1·24-s + 99.0·27-s − 32·32-s + 45.4·33-s + 60.7·34-s + 103.·36-s − 4.60·38-s − 35.7·41-s + 14·43-s + 30.7·44-s + 94.3·48-s + 49·49-s − 179.·51-s − 198.·54-s + ⋯ |
| L(s) = 1 | − 2-s + 1.96·3-s + 4-s − 1.96·6-s − 8-s + 2.86·9-s + 0.699·11-s + 1.96·12-s + 16-s − 1.78·17-s − 2.86·18-s + 0.121·19-s − 0.699·22-s − 1.96·24-s + 3.67·27-s − 32-s + 1.37·33-s + 1.78·34-s + 2.86·36-s − 0.121·38-s − 0.872·41-s + 0.325·43-s + 0.699·44-s + 1.96·48-s + 0.999·49-s − 3.51·51-s − 3.67·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.928999670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.928999670\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5.89T + 9T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 7.69T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 30.3T + 289T^{2} \) |
| 19 | \( 1 - 2.30T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 35.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 14T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 82T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 133.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 100.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 123.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 161.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.27883616942764975746933213364, −10.92184166379660704826162359994, −9.841411892061291453868873827766, −9.022952541985206914955048899807, −8.532723390981625574022178280883, −7.44307623777668857230517091856, −6.61879711955856916405278553124, −4.17703774211329435934034587202, −2.85869565789847382378012851866, −1.71579076340713371229261130083,
1.71579076340713371229261130083, 2.85869565789847382378012851866, 4.17703774211329435934034587202, 6.61879711955856916405278553124, 7.44307623777668857230517091856, 8.532723390981625574022178280883, 9.022952541985206914955048899807, 9.841411892061291453868873827766, 10.92184166379660704826162359994, 12.27883616942764975746933213364
