L(s) = 1 | + 3-s − 0.642·5-s − 2.57·7-s + 9-s − 1.59·11-s − 3.01·13-s − 0.642·15-s + 5.85·17-s + 4.15·19-s − 2.57·21-s + 6.16·23-s − 4.58·25-s + 27-s + 0.244·29-s − 5.80·31-s − 1.59·33-s + 1.65·35-s − 2.35·37-s − 3.01·39-s + 3.00·41-s − 1.02·43-s − 0.642·45-s − 9.29·47-s − 0.359·49-s + 5.85·51-s − 8.77·53-s + 1.02·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.287·5-s − 0.973·7-s + 0.333·9-s − 0.480·11-s − 0.837·13-s − 0.165·15-s + 1.41·17-s + 0.953·19-s − 0.562·21-s + 1.28·23-s − 0.917·25-s + 0.192·27-s + 0.0454·29-s − 1.04·31-s − 0.277·33-s + 0.279·35-s − 0.387·37-s − 0.483·39-s + 0.469·41-s − 0.155·43-s − 0.0957·45-s − 1.35·47-s − 0.0513·49-s + 0.819·51-s − 1.20·53-s + 0.138·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 + T \) |
good | 5 | \( 1 + 0.642T + 5T^{2} \) |
| 7 | \( 1 + 2.57T + 7T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 13 | \( 1 + 3.01T + 13T^{2} \) |
| 17 | \( 1 - 5.85T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 - 0.244T + 29T^{2} \) |
| 31 | \( 1 + 5.80T + 31T^{2} \) |
| 37 | \( 1 + 2.35T + 37T^{2} \) |
| 41 | \( 1 - 3.00T + 41T^{2} \) |
| 43 | \( 1 + 1.02T + 43T^{2} \) |
| 47 | \( 1 + 9.29T + 47T^{2} \) |
| 53 | \( 1 + 8.77T + 53T^{2} \) |
| 59 | \( 1 - 1.90T + 59T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 - 0.795T + 67T^{2} \) |
| 71 | \( 1 - 6.62T + 71T^{2} \) |
| 73 | \( 1 - 5.75T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 5.53T + 83T^{2} \) |
| 89 | \( 1 - 6.54T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.66756315554763890959941899467, −7.21685131662174269055977440048, −6.42140632592927415815028034443, −5.43216411640942093647142641148, −4.95896338798984186334293017285, −3.72937600399808326029392644728, −3.25781804993775580954388633411, −2.56344323193414432664447243098, −1.33199506475897315149074044766, 0,
1.33199506475897315149074044766, 2.56344323193414432664447243098, 3.25781804993775580954388633411, 3.72937600399808326029392644728, 4.95896338798984186334293017285, 5.43216411640942093647142641148, 6.42140632592927415815028034443, 7.21685131662174269055977440048, 7.66756315554763890959941899467