| L(s) = 1 | − 8·5-s − 6·9-s − 8·13-s − 4·17-s + 38·25-s − 8·29-s + 24·37-s − 20·41-s + 48·45-s − 14·49-s − 8·53-s + 24·61-s + 64·65-s − 12·73-s + 27·81-s + 32·85-s + 20·89-s − 36·97-s − 40·101-s − 40·109-s − 28·113-s + 48·117-s − 22·121-s − 136·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3.57·5-s − 2·9-s − 2.21·13-s − 0.970·17-s + 38/5·25-s − 1.48·29-s + 3.94·37-s − 3.12·41-s + 7.15·45-s − 2·49-s − 1.09·53-s + 3.07·61-s + 7.93·65-s − 1.40·73-s + 3·81-s + 3.47·85-s + 2.11·89-s − 3.65·97-s − 3.98·101-s − 3.83·109-s − 2.63·113-s + 4.43·117-s − 2·121-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−14.90732270630889802126855679957, −14.68484230004755975542630052188, −14.68484230004755975542630052188, −13.24730384443706015850945875401, −13.24730384443706015850945875401, −12.21678935249981791272802458671, −12.21678935249981791272802458671, −11.56957687474889255866293555846, −11.56957687474889255866293555846, −10.93612266887391281534482819402, −10.93612266887391281534482819402, −9.531816445825083349512041454382, −9.531816445825083349512041454382, −8.389848986528649416792303765934, −8.389848986528649416792303765934, −7.74661145077041678977550300968, −7.74661145077041678977550300968, −6.71650754317349099241460586550, −6.71650754317349099241460586550, −5.15538859196444031818760258484, −5.15538859196444031818760258484, −4.06135513439308426356140505035, −4.06135513439308426356140505035, −2.84412696583825002808260003434, −2.84412696583825002808260003434, 0, 0,
2.84412696583825002808260003434, 2.84412696583825002808260003434, 4.06135513439308426356140505035, 4.06135513439308426356140505035, 5.15538859196444031818760258484, 5.15538859196444031818760258484, 6.71650754317349099241460586550, 6.71650754317349099241460586550, 7.74661145077041678977550300968, 7.74661145077041678977550300968, 8.389848986528649416792303765934, 8.389848986528649416792303765934, 9.531816445825083349512041454382, 9.531816445825083349512041454382, 10.93612266887391281534482819402, 10.93612266887391281534482819402, 11.56957687474889255866293555846, 11.56957687474889255866293555846, 12.21678935249981791272802458671, 12.21678935249981791272802458671, 13.24730384443706015850945875401, 13.24730384443706015850945875401, 14.68484230004755975542630052188, 14.68484230004755975542630052188, 14.90732270630889802126855679957
