| L(s) = 1 | + 4.54·5-s − 7·7-s − 40.7·11-s + 53.2·13-s − 4.54·17-s − 122.·19-s + 131.·23-s − 104.·25-s + 216.·29-s + 251.·31-s − 31.8·35-s + 11.8·37-s + 111.·41-s − 369.·43-s − 262.·47-s + 49·49-s + 567.·53-s − 185.·55-s + 839.·59-s − 485.·61-s + 242.·65-s + 333.·67-s + 590.·71-s + 490.·73-s + 285.·77-s − 121.·79-s + 609.·83-s + ⋯ |
| L(s) = 1 | + 0.406·5-s − 0.377·7-s − 1.11·11-s + 1.13·13-s − 0.0649·17-s − 1.48·19-s + 1.19·23-s − 0.834·25-s + 1.38·29-s + 1.45·31-s − 0.153·35-s + 0.0528·37-s + 0.425·41-s − 1.30·43-s − 0.815·47-s + 0.142·49-s + 1.46·53-s − 0.454·55-s + 1.85·59-s − 1.01·61-s + 0.462·65-s + 0.608·67-s + 0.986·71-s + 0.786·73-s + 0.422·77-s − 0.173·79-s + 0.806·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.947147244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.947147244\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 4.54T + 125T^{2} \) |
| 11 | \( 1 + 40.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 11.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 839.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 590.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 121.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 637.T + 9.12e5T^{2} \) |
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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
−9.718179127631712603949211309212, −8.538019660001395702464343676051, −8.217417217290693749410373615075, −6.84814280182481527396215659832, −6.25635773208148525417392006906, −5.30092015425988902100910536892, −4.31485113051994028133844482208, −3.13342038055387421546999770604, −2.17058992776661451890067840567, −0.73516376294591578514889153433,
0.73516376294591578514889153433, 2.17058992776661451890067840567, 3.13342038055387421546999770604, 4.31485113051994028133844482208, 5.30092015425988902100910536892, 6.25635773208148525417392006906, 6.84814280182481527396215659832, 8.217417217290693749410373615075, 8.538019660001395702464343676051, 9.718179127631712603949211309212
