L(s) = 1 | − 2·3-s − 4·4-s − 2·5-s + 3·9-s + 8·12-s + 4·15-s + 12·16-s + 8·20-s + 12·23-s + 3·25-s − 4·27-s + 2·31-s − 12·36-s − 10·37-s − 6·45-s − 24·47-s − 24·48-s − 11·49-s + 12·53-s − 16·60-s − 32·64-s − 10·67-s − 24·69-s − 12·71-s − 6·75-s − 24·80-s + 5·81-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 2·4-s − 0.894·5-s + 9-s + 2.30·12-s + 1.03·15-s + 3·16-s + 1.78·20-s + 2.50·23-s + 3/5·25-s − 0.769·27-s + 0.359·31-s − 2·36-s − 1.64·37-s − 0.894·45-s − 3.50·47-s − 3.46·48-s − 1.57·49-s + 1.64·53-s − 2.06·60-s − 4·64-s − 1.22·67-s − 2.88·69-s − 1.42·71-s − 0.692·75-s − 2.68·80-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−8.967386097377301713078020815996, −8.715422139509778262017307101792, −8.312451801137936375363717875728, −8.096375974871344841906660400344, −7.30143486974116417653142659763, −7.29921933322431972523395961305, −6.48616862248567558759622718411, −6.45762518405874774990241488267, −5.47463473472307727305398745504, −5.30679273967680233329666995509, −5.04796426311487769793515776801, −4.55249069074579412970109114024, −4.30206529941477171525949680124, −3.72760061035960114855753063621, −3.23018612043223906164273475939, −2.91860150094527516812052659571, −1.34769574688992792030614946027, −1.25475203166947893171343135556, 0, 0,
1.25475203166947893171343135556, 1.34769574688992792030614946027, 2.91860150094527516812052659571, 3.23018612043223906164273475939, 3.72760061035960114855753063621, 4.30206529941477171525949680124, 4.55249069074579412970109114024, 5.04796426311487769793515776801, 5.30679273967680233329666995509, 5.47463473472307727305398745504, 6.45762518405874774990241488267, 6.48616862248567558759622718411, 7.29921933322431972523395961305, 7.30143486974116417653142659763, 8.096375974871344841906660400344, 8.312451801137936375363717875728, 8.715422139509778262017307101792, 8.967386097377301713078020815996