| L(s) = 1 | + 4.47·3-s − 31.3·7-s − 6.99·9-s + 8.94·11-s + 62·13-s + 46·17-s − 107.·19-s − 140·21-s + 192.·23-s − 152.·27-s − 90·29-s + 152.·31-s + 40.0·33-s + 214·37-s + 277.·39-s − 10·41-s − 67.0·43-s + 398.·47-s + 637.·49-s + 205.·51-s + 678·53-s − 480.·57-s + 411.·59-s + 250·61-s + 219.·63-s + 49.1·67-s + 860·69-s + ⋯ |
| L(s) = 1 | + 0.860·3-s − 1.69·7-s − 0.259·9-s + 0.245·11-s + 1.32·13-s + 0.656·17-s − 1.29·19-s − 1.45·21-s + 1.74·23-s − 1.08·27-s − 0.576·29-s + 0.880·31-s + 0.211·33-s + 0.950·37-s + 1.13·39-s − 0.0380·41-s − 0.237·43-s + 1.23·47-s + 1.85·49-s + 0.564·51-s + 1.75·53-s − 1.11·57-s + 0.907·59-s + 0.524·61-s + 0.438·63-s + 0.0897·67-s + 1.50·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.228255867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.228255867\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4.47T + 27T^{2} \) |
| 7 | \( 1 + 31.3T + 343T^{2} \) |
| 11 | \( 1 - 8.94T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 214T + 5.06e4T^{2} \) |
| 41 | \( 1 + 10T + 6.89e4T^{2} \) |
| 43 | \( 1 + 67.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 678T + 1.48e5T^{2} \) |
| 59 | \( 1 - 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 250T + 2.26e5T^{2} \) |
| 67 | \( 1 - 49.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 366.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 522T + 3.89e5T^{2} \) |
| 79 | \( 1 + 876.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 970T + 7.04e5T^{2} \) |
| 97 | \( 1 - 934T + 9.12e5T^{2} \) |
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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
−9.713870029585675179267844762474, −8.909487792563950965334600984111, −8.497475019525728120468287910594, −7.24925138099684314018904573705, −6.39787878816246666706931039731, −5.68601423791891021210630987358, −4.02786195530899935473202259176, −3.32103381286621338463473500847, −2.49793498374524678572356945045, −0.794067071034629511385094427932,
0.794067071034629511385094427932, 2.49793498374524678572356945045, 3.32103381286621338463473500847, 4.02786195530899935473202259176, 5.68601423791891021210630987358, 6.39787878816246666706931039731, 7.24925138099684314018904573705, 8.497475019525728120468287910594, 8.909487792563950965334600984111, 9.713870029585675179267844762474
