| 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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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
−9.843793468558168467740768141180, −8.791873130240302564114347326178, −7.25614478151514303450426943562, −6.59947875026069205621554412768, −6.05905876754360656004394905071, −5.15282062620040149458430425516, −4.11713957347578463943351156022, −2.89194778919852650489326717125, −0.979881847225025387591293291526, 0,
0.979881847225025387591293291526, 2.89194778919852650489326717125, 4.11713957347578463943351156022, 5.15282062620040149458430425516, 6.05905876754360656004394905071, 6.59947875026069205621554412768, 7.25614478151514303450426943562, 8.791873130240302564114347326178, 9.843793468558168467740768141180
