| L(s) = 1 | + 4.47·3-s − 5·5-s + 13.4·7-s + 11.0·9-s − 22.3·15-s + 60.0·21-s + 13.4·23-s + 25·25-s + 8.94·27-s + 22·29-s − 67.0·35-s − 62·41-s + 40.2·43-s − 55.0·45-s − 93.9·47-s + 131.·49-s − 58·61-s + 147.·63-s − 67.0·67-s + 60.0·69-s + 111.·75-s − 58.9·81-s + 147.·83-s + 98.3·87-s − 142·89-s − 122·101-s − 201.·103-s + ⋯ |
| L(s) = 1 | + 1.49·3-s − 5-s + 1.91·7-s + 1.22·9-s − 1.49·15-s + 2.85·21-s + 0.583·23-s + 25-s + 0.331·27-s + 0.758·29-s − 1.91·35-s − 1.51·41-s + 0.936·43-s − 1.22·45-s − 1.99·47-s + 2.67·49-s − 0.950·61-s + 2.34·63-s − 1.00·67-s + 0.869·69-s + 1.49·75-s − 0.728·81-s + 1.77·83-s + 1.13·87-s − 1.59·89-s − 1.20·101-s − 1.95·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.735925465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.735925465\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 3 | \( 1 - 4.47T + 9T^{2} \) |
| 7 | \( 1 - 13.4T + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 13.4T + 529T^{2} \) |
| 29 | \( 1 - 22T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 62T + 1.68e3T^{2} \) |
| 43 | \( 1 - 40.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 93.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 58T + 3.72e3T^{2} \) |
| 67 | \( 1 + 67.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 147.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 142T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−11.42028100071570442448550288373, −10.58396912500989155239501107742, −9.192599903436878639929325733107, −8.272494182758512211882887558773, −8.024531733774726137243229682118, −7.04409871767600335271536095341, −5.02171727748599603414367998988, −4.15755819641156367939744542741, −2.94469763204670039766568168655, −1.55343992460716092921524765545,
1.55343992460716092921524765545, 2.94469763204670039766568168655, 4.15755819641156367939744542741, 5.02171727748599603414367998988, 7.04409871767600335271536095341, 8.024531733774726137243229682118, 8.272494182758512211882887558773, 9.192599903436878639929325733107, 10.58396912500989155239501107742, 11.42028100071570442448550288373
