| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 5·9-s − 4·14-s + 16-s + 16·17-s − 5·18-s + 8·23-s − 9·25-s − 4·28-s − 6·31-s + 32-s + 16·34-s − 5·36-s + 4·41-s + 8·46-s + 26·47-s − 2·49-s − 9·50-s − 4·56-s − 6·62-s + 20·63-s + 64-s + 16·68-s + 4·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 5/3·9-s − 1.06·14-s + 1/4·16-s + 3.88·17-s − 1.17·18-s + 1.66·23-s − 9/5·25-s − 0.755·28-s − 1.07·31-s + 0.176·32-s + 2.74·34-s − 5/6·36-s + 0.624·41-s + 1.17·46-s + 3.79·47-s − 2/7·49-s − 1.27·50-s − 0.534·56-s − 0.762·62-s + 2.51·63-s + 1/8·64-s + 1.94·68-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107648 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107648 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.887122613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.887122613\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−9.633495492381883994739439375568, −9.187696200797261240168787461144, −8.538714587534864706883628474190, −7.71436185563727090632996745083, −7.68992355213504990402216682341, −7.02533007960674119090827223190, −6.11658999858863662469236119804, −5.97874250935484417499511635389, −5.35705266569945906014064710560, −5.25136790623537531107690201816, −3.80259981482385994331528030455, −3.53944000086915839540994955829, −3.07071881135695746551942332895, −2.44287708620897533980700790337, −0.907460307275556682567938804358,
0.907460307275556682567938804358, 2.44287708620897533980700790337, 3.07071881135695746551942332895, 3.53944000086915839540994955829, 3.80259981482385994331528030455, 5.25136790623537531107690201816, 5.35705266569945906014064710560, 5.97874250935484417499511635389, 6.11658999858863662469236119804, 7.02533007960674119090827223190, 7.68992355213504990402216682341, 7.71436185563727090632996745083, 8.538714587534864706883628474190, 9.187696200797261240168787461144, 9.633495492381883994739439375568
