| L(s) = 1 | + 4-s − 6·5-s + 16-s − 6·20-s + 17·25-s − 8·31-s − 2·37-s − 24·47-s + 2·49-s − 12·53-s + 64-s − 8·67-s − 24·71-s − 6·80-s − 6·89-s + 4·97-s + 17·100-s − 8·103-s − 30·113-s − 11·121-s − 8·124-s − 18·125-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2.68·5-s + 1/4·16-s − 1.34·20-s + 17/5·25-s − 1.43·31-s − 0.328·37-s − 3.50·47-s + 2/7·49-s − 1.64·53-s + 1/8·64-s − 0.977·67-s − 2.84·71-s − 0.670·80-s − 0.635·89-s + 0.406·97-s + 1.69·100-s − 0.788·103-s − 2.82·113-s − 121-s − 0.718·124-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.164·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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
−7.16277862972928710155421440227, −6.98504291328603819496217788978, −6.29261335708839240986736404399, −6.07141567781617112259653788579, −5.16414682280697373873566197697, −5.04773425377657589958068791936, −4.25638665319375785757127060430, −4.17461543307518243917716468023, −3.57784119065864589215915440592, −3.12147196896030897278593721655, −2.94651413139897301734292877776, −1.84842910920498472471311986628, −1.34591687981300633859944824892, 0, 0,
1.34591687981300633859944824892, 1.84842910920498472471311986628, 2.94651413139897301734292877776, 3.12147196896030897278593721655, 3.57784119065864589215915440592, 4.17461543307518243917716468023, 4.25638665319375785757127060430, 5.04773425377657589958068791936, 5.16414682280697373873566197697, 6.07141567781617112259653788579, 6.29261335708839240986736404399, 6.98504291328603819496217788978, 7.16277862972928710155421440227
