| L(s) = 1 | + 3-s + 4-s + 4·7-s + 9-s + 12-s + 2·13-s + 16-s + 4·19-s + 4·21-s + 25-s + 27-s + 4·28-s − 8·31-s + 36-s + 4·37-s + 2·39-s − 8·43-s + 48-s − 2·49-s + 2·52-s + 4·57-s − 20·61-s + 4·63-s + 64-s + 16·67-s + 16·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.51·7-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 0.755·28-s − 1.43·31-s + 1/6·36-s + 0.657·37-s + 0.320·39-s − 1.21·43-s + 0.144·48-s − 2/7·49-s + 0.277·52-s + 0.529·57-s − 2.56·61-s + 0.503·63-s + 1/8·64-s + 1.95·67-s + 1.87·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.557653723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.557653723\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.281210776999896926069171305774, −8.195553408380404831920993641146, −7.69992577245459400562531294135, −7.44200053996330116321929479527, −6.58298325388265669739783786821, −6.56327661741499604036243526620, −5.59531005218261041497813692032, −5.30751325359973115442660542578, −4.75705494829356928996463504748, −4.27346273235141665749062775801, −3.39519945126596367924943215518, −3.28257229246751367368737648325, −2.19648591121885948588699453544, −1.81386816189337101345457960761, −1.07994497826633863377468377161,
1.07994497826633863377468377161, 1.81386816189337101345457960761, 2.19648591121885948588699453544, 3.28257229246751367368737648325, 3.39519945126596367924943215518, 4.27346273235141665749062775801, 4.75705494829356928996463504748, 5.30751325359973115442660542578, 5.59531005218261041497813692032, 6.56327661741499604036243526620, 6.58298325388265669739783786821, 7.44200053996330116321929479527, 7.69992577245459400562531294135, 8.195553408380404831920993641146, 8.281210776999896926069171305774
