| L(s) = 1 | + 9.23·3-s + 33.5·7-s + 58.3·9-s + 310.·21-s − 73.8·23-s + 289.·27-s + 306·29-s − 460.·41-s − 563.·43-s − 41.1·47-s + 785.·49-s − 40.2·61-s + 1.95e3·63-s + 1.16·67-s − 682.·69-s + 1.09e3·81-s + 989.·83-s + 2.82e3·87-s − 1.38e3·89-s − 378·101-s + 663.·103-s − 1.32e3·107-s + 1.97e3·109-s + ⋯ |
| L(s) = 1 | + 1.77·3-s + 1.81·7-s + 2.15·9-s + 3.22·21-s − 0.669·23-s + 2.06·27-s + 1.95·29-s − 1.75·41-s − 1.99·43-s − 0.127·47-s + 2.29·49-s − 0.0844·61-s + 3.91·63-s + 0.00212·67-s − 1.19·69-s + 1.50·81-s + 1.30·83-s + 3.48·87-s − 1.65·89-s − 0.372·101-s + 0.634·103-s − 1.20·107-s + 1.73·109-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
| \(L(2)\) |
\(\approx\) |
\(5.124806794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.124806794\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 9.23T + 27T^{2} \) |
| 7 | \( 1 - 33.5T + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 306T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 + 460.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 563.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 41.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 40.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.16T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 - 989.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−9.808609048281878972578923981327, −8.557372398874795494845711482973, −8.410943102383548587749600420653, −7.67237284512240269463608962625, −6.70643146102231395000209422300, −5.07118577408370535203430362419, −4.35067748268439510042753276613, −3.26052586723759459267075886264, −2.15707497683832617851259185625, −1.38649674374557187296766448970,
1.38649674374557187296766448970, 2.15707497683832617851259185625, 3.26052586723759459267075886264, 4.35067748268439510042753276613, 5.07118577408370535203430362419, 6.70643146102231395000209422300, 7.67237284512240269463608962625, 8.410943102383548587749600420653, 8.557372398874795494845711482973, 9.808609048281878972578923981327
