| L(s) = 1 | − 6·5-s − 20·7-s + 24·11-s + 13·13-s + 30·17-s + 16·19-s − 72·23-s − 89·25-s + 282·29-s − 164·31-s + 120·35-s + 110·37-s + 126·41-s − 164·43-s − 204·47-s + 57·49-s + 738·53-s − 144·55-s + 120·59-s + 614·61-s − 78·65-s − 848·67-s + 132·71-s + 218·73-s − 480·77-s + 1.09e3·79-s + 552·83-s + ⋯ |
| L(s) = 1 | − 0.536·5-s − 1.07·7-s + 0.657·11-s + 0.277·13-s + 0.428·17-s + 0.193·19-s − 0.652·23-s − 0.711·25-s + 1.80·29-s − 0.950·31-s + 0.579·35-s + 0.488·37-s + 0.479·41-s − 0.581·43-s − 0.633·47-s + 0.166·49-s + 1.91·53-s − 0.353·55-s + 0.264·59-s + 1.28·61-s − 0.148·65-s − 1.54·67-s + 0.220·71-s + 0.349·73-s − 0.710·77-s + 1.56·79-s + 0.729·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 16 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 282 T + p^{3} T^{2} \) |
| 31 | \( 1 + 164 T + p^{3} T^{2} \) |
| 37 | \( 1 - 110 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 204 T + p^{3} T^{2} \) |
| 53 | \( 1 - 738 T + p^{3} T^{2} \) |
| 59 | \( 1 - 120 T + p^{3} T^{2} \) |
| 61 | \( 1 - 614 T + p^{3} T^{2} \) |
| 67 | \( 1 + 848 T + p^{3} T^{2} \) |
| 71 | \( 1 - 132 T + p^{3} T^{2} \) |
| 73 | \( 1 - 218 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1096 T + p^{3} T^{2} \) |
| 83 | \( 1 - 552 T + p^{3} T^{2} \) |
| 89 | \( 1 + 210 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1726 T + p^{3} T^{2} \) |
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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
−8.460927541068129522516634245310, −7.72767882584697136002012332561, −6.79043842710023797969185955056, −6.23989418328161810465906319080, −5.29851559016372706222394964015, −4.08389471868059630471562959481, −3.56478878686693924459734333556, −2.53297029940447166104542452433, −1.13002115081879362538873988443, 0,
1.13002115081879362538873988443, 2.53297029940447166104542452433, 3.56478878686693924459734333556, 4.08389471868059630471562959481, 5.29851559016372706222394964015, 6.23989418328161810465906319080, 6.79043842710023797969185955056, 7.72767882584697136002012332561, 8.460927541068129522516634245310
