| L(s) = 1 | − 4-s − 3·9-s + 13-s + 16-s − 5·17-s + 4·23-s + 2·25-s + 3·29-s + 3·36-s − 15·43-s − 49-s − 52-s + 5·53-s + 6·61-s − 64-s + 5·68-s + 13·79-s + 9·81-s − 4·92-s − 2·100-s − 23·101-s + 9·103-s − 14·107-s − 13·113-s − 3·116-s − 3·117-s − 5·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 9-s + 0.277·13-s + 1/4·16-s − 1.21·17-s + 0.834·23-s + 2/5·25-s + 0.557·29-s + 1/2·36-s − 2.28·43-s − 1/7·49-s − 0.138·52-s + 0.686·53-s + 0.768·61-s − 1/8·64-s + 0.606·68-s + 1.46·79-s + 81-s − 0.417·92-s − 1/5·100-s − 2.28·101-s + 0.886·103-s − 1.35·107-s − 1.22·113-s − 0.278·116-s − 0.277·117-s − 0.454·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 57 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 125 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.527729637139246442970547514289, −8.361819412769653396085952898581, −7.81373180726234109768967619040, −7.11623468106399549159110730427, −6.56005282084381892752593328885, −6.42446513930398993452890650537, −5.61696812952213605734643195676, −5.06837807019022937183146889910, −4.87462272731706901675469786264, −4.04282946894073052755253663592, −3.55541633401756244992512767468, −2.85581817800940167802022404831, −2.30535224266903704554955416039, −1.23471090157141841961253097684, 0,
1.23471090157141841961253097684, 2.30535224266903704554955416039, 2.85581817800940167802022404831, 3.55541633401756244992512767468, 4.04282946894073052755253663592, 4.87462272731706901675469786264, 5.06837807019022937183146889910, 5.61696812952213605734643195676, 6.42446513930398993452890650537, 6.56005282084381892752593328885, 7.11623468106399549159110730427, 7.81373180726234109768967619040, 8.361819412769653396085952898581, 8.527729637139246442970547514289
