| L(s) = 1 | − 2·4-s − 7-s + 6·13-s + 4·16-s + 5·17-s + 19-s − 3·23-s + 2·28-s + 3·29-s + 2·31-s − 10·37-s + 4·41-s − 9·43-s − 8·47-s + 49-s − 12·52-s − 53-s + 5·59-s − 7·61-s − 8·64-s − 2·67-s − 10·68-s + 2·73-s − 2·76-s + 12·79-s − 9·83-s + 9·89-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 1.66·13-s + 16-s + 1.21·17-s + 0.229·19-s − 0.625·23-s + 0.377·28-s + 0.557·29-s + 0.359·31-s − 1.64·37-s + 0.624·41-s − 1.37·43-s − 1.16·47-s + 1/7·49-s − 1.66·52-s − 0.137·53-s + 0.650·59-s − 0.896·61-s − 64-s − 0.244·67-s − 1.21·68-s + 0.234·73-s − 0.229·76-s + 1.35·79-s − 0.987·83-s + 0.953·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−13.43677079221417, −12.96160136771828, −12.39408994832262, −12.05683716983510, −11.55609060194399, −10.87539203762844, −10.46731745672560, −9.861649323966389, −9.758810582363023, −8.970553389648145, −8.575096224798476, −8.241225616180638, −7.749692173350621, −7.122144189656329, −6.386215006117606, −6.106550230406189, −5.501353578082409, −5.049566880659734, −4.468017577660084, −3.815878581581588, −3.314565878195043, −3.195702879962067, −2.012377139908587, −1.346938405269013, −0.8342484944170100, 0,
0.8342484944170100, 1.346938405269013, 2.012377139908587, 3.195702879962067, 3.314565878195043, 3.815878581581588, 4.468017577660084, 5.049566880659734, 5.501353578082409, 6.106550230406189, 6.386215006117606, 7.122144189656329, 7.749692173350621, 8.241225616180638, 8.575096224798476, 8.970553389648145, 9.758810582363023, 9.861649323966389, 10.46731745672560, 10.87539203762844, 11.55609060194399, 12.05683716983510, 12.39408994832262, 12.96160136771828, 13.43677079221417
