| L(s) = 1 | − 2·3-s + 12.1·5-s − 7·7-s − 23·9-s + 11·11-s − 16.3·13-s − 24.3·15-s + 30.4·17-s + 43.1·19-s + 14·21-s + 138.·23-s + 22.9·25-s + 100·27-s − 18.9·29-s + 20.1·31-s − 22·33-s − 85.1·35-s + 83.3·37-s + 32.6·39-s − 277.·41-s + 379.·43-s − 279.·45-s + 137.·47-s + 49·49-s − 60.9·51-s + 6.99·53-s + 133.·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 1.08·5-s − 0.377·7-s − 0.851·9-s + 0.301·11-s − 0.348·13-s − 0.418·15-s + 0.435·17-s + 0.521·19-s + 0.145·21-s + 1.25·23-s + 0.183·25-s + 0.712·27-s − 0.121·29-s + 0.116·31-s − 0.116·33-s − 0.411·35-s + 0.370·37-s + 0.134·39-s − 1.05·41-s + 1.34·43-s − 0.926·45-s + 0.425·47-s + 0.142·49-s − 0.167·51-s + 0.0181·53-s + 0.328·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.906702411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.906702411\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 2T + 27T^{2} \) |
| 5 | \( 1 - 12.1T + 125T^{2} \) |
| 13 | \( 1 + 16.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 30.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 18.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 20.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 83.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 277.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 379.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 137.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 6.99T + 1.48e5T^{2} \) |
| 59 | \( 1 + 160.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 499.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 319.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 699.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 181.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 640.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 617.T + 9.12e5T^{2} \) |
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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
−10.15121835027089500990482766180, −9.425837093179689911178496907344, −8.698556021186746690859352456922, −7.43996449460838653420156295329, −6.43619475930488780151446256329, −5.70388581418520990994110108970, −4.95020085092151565868102830871, −3.37347017909953614712346188513, −2.31323576181308357441243259491, −0.840144098044661097349100324063,
0.840144098044661097349100324063, 2.31323576181308357441243259491, 3.37347017909953614712346188513, 4.95020085092151565868102830871, 5.70388581418520990994110108970, 6.43619475930488780151446256329, 7.43996449460838653420156295329, 8.698556021186746690859352456922, 9.425837093179689911178496907344, 10.15121835027089500990482766180
