| L(s) = 1 | − 4-s − 2·11-s + 16-s + 8·19-s − 20·29-s − 4·41-s + 2·44-s + 14·49-s − 8·59-s − 4·61-s − 64-s + 16·71-s − 8·76-s + 16·79-s − 12·89-s − 28·101-s − 28·109-s + 20·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.603·11-s + 1/4·16-s + 1.83·19-s − 3.71·29-s − 0.624·41-s + 0.301·44-s + 2·49-s − 1.04·59-s − 0.512·61-s − 1/8·64-s + 1.89·71-s − 0.917·76-s + 1.80·79-s − 1.27·89-s − 2.78·101-s − 2.68·109-s + 1.85·116-s + 3/11·121-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 + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4702316182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4702316182\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.670563762419705587139263403963, −7.80621756321457333464519571698, −7.77173886092578000657023967578, −7.41557009097787577091059953207, −7.23663214637682138777299255441, −6.59811478441517563200216281847, −6.26745913427216178063214876511, −5.75641638655036865906530279548, −5.35124486263614692992689111903, −5.18632533652148509120793466199, −5.07103594796364094799336229336, −4.18897595931298884188577851589, −3.83156702845187141458932196008, −3.69234893575818477385860524501, −3.19597223770976866348755001487, −2.44455977518418987463763315738, −2.40129407964185499217871152780, −1.40147386139176919626478970156, −1.27350617475979543111695547732, −0.19308061830019867957196254470,
0.19308061830019867957196254470, 1.27350617475979543111695547732, 1.40147386139176919626478970156, 2.40129407964185499217871152780, 2.44455977518418987463763315738, 3.19597223770976866348755001487, 3.69234893575818477385860524501, 3.83156702845187141458932196008, 4.18897595931298884188577851589, 5.07103594796364094799336229336, 5.18632533652148509120793466199, 5.35124486263614692992689111903, 5.75641638655036865906530279548, 6.26745913427216178063214876511, 6.59811478441517563200216281847, 7.23663214637682138777299255441, 7.41557009097787577091059953207, 7.77173886092578000657023967578, 7.80621756321457333464519571698, 8.670563762419705587139263403963
