| L(s) = 1 | + 3·3-s − 11·5-s + 9·9-s − 39·11-s − 32·13-s − 33·15-s + 12·17-s + 88·19-s + 92·23-s − 4·25-s + 27·27-s + 255·29-s + 35·31-s − 117·33-s − 4·37-s − 96·39-s + 16·41-s + 330·43-s − 99·45-s + 298·47-s + 36·51-s − 717·53-s + 429·55-s + 264·57-s + 217·59-s + 386·61-s + 352·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.983·5-s + 1/3·9-s − 1.06·11-s − 0.682·13-s − 0.568·15-s + 0.171·17-s + 1.06·19-s + 0.834·23-s − 0.0319·25-s + 0.192·27-s + 1.63·29-s + 0.202·31-s − 0.617·33-s − 0.0177·37-s − 0.394·39-s + 0.0609·41-s + 1.17·43-s − 0.327·45-s + 0.924·47-s + 0.0988·51-s − 1.85·53-s + 1.05·55-s + 0.613·57-s + 0.478·59-s + 0.810·61-s + 0.671·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 11 T + p^{3} T^{2} \) |
| 11 | \( 1 + 39 T + p^{3} T^{2} \) |
| 13 | \( 1 + 32 T + p^{3} T^{2} \) |
| 17 | \( 1 - 12 T + p^{3} T^{2} \) |
| 19 | \( 1 - 88 T + p^{3} T^{2} \) |
| 23 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 255 T + p^{3} T^{2} \) |
| 31 | \( 1 - 35 T + p^{3} T^{2} \) |
| 37 | \( 1 + 4 T + p^{3} T^{2} \) |
| 41 | \( 1 - 16 T + p^{3} T^{2} \) |
| 43 | \( 1 - 330 T + p^{3} T^{2} \) |
| 47 | \( 1 - 298 T + p^{3} T^{2} \) |
| 53 | \( 1 + 717 T + p^{3} T^{2} \) |
| 59 | \( 1 - 217 T + p^{3} T^{2} \) |
| 61 | \( 1 - 386 T + p^{3} T^{2} \) |
| 67 | \( 1 + 906 T + p^{3} T^{2} \) |
| 71 | \( 1 - 34 T + p^{3} T^{2} \) |
| 73 | \( 1 + 838 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1325 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1163 T + p^{3} T^{2} \) |
| 89 | \( 1 + 54 T + p^{3} T^{2} \) |
| 97 | \( 1 - 7 T + p^{3} T^{2} \) |
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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
−8.159464019602492905072710058213, −7.51877449185231840143204982098, −7.09313721715660353857067021796, −5.82339828484456049984958451934, −4.89535317209725444383535212438, −4.23665567517680344233195906670, −3.09852154544451988057167191133, −2.64958209440030907931031677119, −1.15454793729519640680553043144, 0,
1.15454793729519640680553043144, 2.64958209440030907931031677119, 3.09852154544451988057167191133, 4.23665567517680344233195906670, 4.89535317209725444383535212438, 5.82339828484456049984958451934, 7.09313721715660353857067021796, 7.51877449185231840143204982098, 8.159464019602492905072710058213
