| L(s) = 1 | + 2·5-s − 6·9-s − 25-s + 20·29-s + 20·41-s − 12·45-s − 14·49-s + 20·61-s + 27·81-s + 20·89-s + 4·101-s − 12·109-s − 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 40·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s − 1/5·25-s + 3.71·29-s + 3.12·41-s − 1.78·45-s − 2·49-s + 2.56·61-s + 3·81-s + 2.11·89-s + 0.398·101-s − 1.14·109-s − 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.537338284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.537338284\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.317356556158666156286958675354, −9.229738681555639603454604612144, −8.548239184977679663193402766242, −8.095900450392739090116843696578, −7.86574055998535746161422058704, −6.83957021602347792551015258406, −6.27282080646876620831649653268, −6.17090310700109824442559838984, −5.42644460572577514350636284077, −5.01452297596172644651029655270, −4.32056015764801105587344203662, −3.38983172672051634596756490050, −2.63895104553179739845027310173, −2.40245494881792658011192485726, −0.923744794836951703848041902340,
0.923744794836951703848041902340, 2.40245494881792658011192485726, 2.63895104553179739845027310173, 3.38983172672051634596756490050, 4.32056015764801105587344203662, 5.01452297596172644651029655270, 5.42644460572577514350636284077, 6.17090310700109824442559838984, 6.27282080646876620831649653268, 6.83957021602347792551015258406, 7.86574055998535746161422058704, 8.095900450392739090116843696578, 8.548239184977679663193402766242, 9.229738681555639603454604612144, 9.317356556158666156286958675354
