| L(s) = 1 | − 4-s − 9-s + 2·11-s + 16-s + 10·19-s − 10·29-s + 14·31-s + 36-s + 4·41-s − 2·44-s + 10·49-s + 20·59-s + 4·61-s − 64-s − 26·71-s − 10·76-s + 81-s + 30·89-s − 2·99-s − 6·101-s − 10·109-s + 10·116-s + 3·121-s − 14·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 0.603·11-s + 1/4·16-s + 2.29·19-s − 1.85·29-s + 2.51·31-s + 1/6·36-s + 0.624·41-s − 0.301·44-s + 10/7·49-s + 2.60·59-s + 0.512·61-s − 1/8·64-s − 3.08·71-s − 1.14·76-s + 1/9·81-s + 3.17·89-s − 0.201·99-s − 0.597·101-s − 0.957·109-s + 0.928·116-s + 3/11·121-s − 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.363298050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.363298050\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−9.512819222213884290545226005859, −9.056701465613106260057737042800, −9.007118018303236241790846607409, −8.402487785780673195710707636634, −7.947821940614761199126848489241, −7.65282270890592114735498203060, −7.20100086731875770690407773355, −6.87044492350716174712725865006, −6.29892024535081578405332832023, −5.82586612805657392075702145599, −5.41290013602389877203645402290, −5.26286478611146894684923725117, −4.39613384935664857116790055038, −4.29152968189868631246558929654, −3.57022832712585150401080758638, −3.22890244243244251918899502031, −2.67075294972446485954149170581, −2.03072796841305413360174752835, −1.14125219965093803274831253406, −0.71192227725883991759629333763,
0.71192227725883991759629333763, 1.14125219965093803274831253406, 2.03072796841305413360174752835, 2.67075294972446485954149170581, 3.22890244243244251918899502031, 3.57022832712585150401080758638, 4.29152968189868631246558929654, 4.39613384935664857116790055038, 5.26286478611146894684923725117, 5.41290013602389877203645402290, 5.82586612805657392075702145599, 6.29892024535081578405332832023, 6.87044492350716174712725865006, 7.20100086731875770690407773355, 7.65282270890592114735498203060, 7.947821940614761199126848489241, 8.402487785780673195710707636634, 9.007118018303236241790846607409, 9.056701465613106260057737042800, 9.512819222213884290545226005859
