| L(s) = 1 | − 15.3·5-s + 7·7-s − 5.35·11-s + 11.2·13-s − 94.0·17-s − 20·19-s + 102.·23-s + 110.·25-s − 102·29-s − 341.·31-s − 107.·35-s + 288.·37-s + 252.·41-s + 145.·43-s − 573.·47-s + 49·49-s + 234.·53-s + 82.2·55-s − 151.·59-s + 243.·61-s − 173.·65-s − 142.·67-s − 65.0·71-s + 380.·73-s − 37.5·77-s + 830.·79-s + 469.·83-s + ⋯ |
| L(s) = 1 | − 1.37·5-s + 0.377·7-s − 0.146·11-s + 0.240·13-s − 1.34·17-s − 0.241·19-s + 0.931·23-s + 0.886·25-s − 0.653·29-s − 1.97·31-s − 0.519·35-s + 1.28·37-s + 0.963·41-s + 0.516·43-s − 1.78·47-s + 0.142·49-s + 0.607·53-s + 0.201·55-s − 0.333·59-s + 0.511·61-s − 0.330·65-s − 0.260·67-s − 0.108·71-s + 0.609·73-s − 0.0555·77-s + 1.18·79-s + 0.620·83-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
| \(L(2)\) |
\(\approx\) |
\(1.144628567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.144628567\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 15.3T + 125T^{2} \) |
| 11 | \( 1 + 5.35T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20T + 6.85e3T^{2} \) |
| 23 | \( 1 - 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 102T + 2.43e4T^{2} \) |
| 31 | \( 1 + 341.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 288.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 145.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 573.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 151.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 243.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 142.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 65.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 380.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 830.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 469.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.55e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−9.354490921483911820877193861610, −8.734042537551167477848563354100, −7.81899958877800443829918520712, −7.28280335887701988043701500534, −6.27284192562951817248957077995, −5.04686267473103459971677016938, −4.23282611378430671087485866557, −3.42676013913222974215248224201, −2.09889146547322669420512611561, −0.55558228725423504025157534674,
0.55558228725423504025157534674, 2.09889146547322669420512611561, 3.42676013913222974215248224201, 4.23282611378430671087485866557, 5.04686267473103459971677016938, 6.27284192562951817248957077995, 7.28280335887701988043701500534, 7.81899958877800443829918520712, 8.734042537551167477848563354100, 9.354490921483911820877193861610
