L(s) = 1 | − 4·7-s + 4·9-s + 4·17-s − 12·23-s + 8·25-s + 16·31-s + 16·47-s − 2·49-s − 16·63-s − 20·71-s − 8·73-s + 7·81-s − 8·89-s − 4·97-s + 12·103-s + 12·113-s − 16·119-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + 157-s + 48·161-s + ⋯ |
L(s) = 1 | − 1.51·7-s + 4/3·9-s + 0.970·17-s − 2.50·23-s + 8/5·25-s + 2.87·31-s + 2.33·47-s − 2/7·49-s − 2.01·63-s − 2.37·71-s − 0.936·73-s + 7/9·81-s − 0.847·89-s − 0.406·97-s + 1.18·103-s + 1.12·113-s − 1.46·119-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.29·153-s + 0.0798·157-s + 3.78·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.850636083\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.850636083\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 84 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 164 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.992676442534183611680202937412, −9.948975380079506745966139116837, −9.547914153327182136731585318217, −8.832670494873253177321195957629, −8.655161200073585778377237657365, −7.899411765043100827645955048993, −7.74883236945244734105833108280, −7.14775107391726688352507500281, −6.70344858586590378313709496923, −6.46078746563502339471693998299, −5.80543070102939722977492335971, −5.74531690973983054508349506842, −4.72812929748733416768286989422, −4.33846451309494687477940766460, −4.10166462469460812271823003049, −3.19994371502056489193105609228, −3.04721980727381412786029043622, −2.27686264849340236244693375021, −1.42605941539100853309724913155, −0.66311245126504812302628570019,
0.66311245126504812302628570019, 1.42605941539100853309724913155, 2.27686264849340236244693375021, 3.04721980727381412786029043622, 3.19994371502056489193105609228, 4.10166462469460812271823003049, 4.33846451309494687477940766460, 4.72812929748733416768286989422, 5.74531690973983054508349506842, 5.80543070102939722977492335971, 6.46078746563502339471693998299, 6.70344858586590378313709496923, 7.14775107391726688352507500281, 7.74883236945244734105833108280, 7.899411765043100827645955048993, 8.655161200073585778377237657365, 8.832670494873253177321195957629, 9.547914153327182136731585318217, 9.948975380079506745966139116837, 9.992676442534183611680202937412