| L(s) = 1 | − 3·3-s − 15.6·5-s + 20.4·7-s + 9·9-s − 44.7·11-s − 13·13-s + 47.0·15-s + 53.4·17-s + 31.9·19-s − 61.3·21-s − 82.4·23-s + 121.·25-s − 27·27-s − 100.·29-s + 263.·31-s + 134.·33-s − 320.·35-s − 104.·37-s + 39·39-s − 398.·41-s + 180.·43-s − 141.·45-s + 118.·47-s + 74.8·49-s − 160.·51-s + 526.·53-s + 702.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.40·5-s + 1.10·7-s + 0.333·9-s − 1.22·11-s − 0.277·13-s + 0.810·15-s + 0.762·17-s + 0.386·19-s − 0.637·21-s − 0.747·23-s + 0.971·25-s − 0.192·27-s − 0.645·29-s + 1.52·31-s + 0.708·33-s − 1.54·35-s − 0.465·37-s + 0.160·39-s − 1.51·41-s + 0.639·43-s − 0.468·45-s + 0.366·47-s + 0.218·49-s − 0.440·51-s + 1.36·53-s + 1.72·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 15.6T + 125T^{2} \) |
| 7 | \( 1 - 20.4T + 343T^{2} \) |
| 11 | \( 1 + 44.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 53.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 82.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 104.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 398.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 180.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 118.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 526.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 224.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 256.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 494.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 146.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 164.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 305.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 208.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 284.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 104.T + 9.12e5T^{2} \) |
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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
−7.967000652747144134792024414971, −7.66357184977265317134384166259, −6.83906518884710769904195681147, −5.58735804321054376861330720300, −5.04955136114577964725672372202, −4.29084781997712664490607394365, −3.43143449422674844543071570008, −2.25832712871942802141479497141, −0.964069540786471249878486171385, 0,
0.964069540786471249878486171385, 2.25832712871942802141479497141, 3.43143449422674844543071570008, 4.29084781997712664490607394365, 5.04955136114577964725672372202, 5.58735804321054376861330720300, 6.83906518884710769904195681147, 7.66357184977265317134384166259, 7.967000652747144134792024414971
