| L(s) = 1 | + 4·11-s + 10·19-s − 16·29-s + 14·31-s + 12·41-s − 2·49-s − 8·59-s + 26·61-s − 12·71-s − 18·79-s + 12·89-s + 24·101-s − 34·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 2.29·19-s − 2.97·29-s + 2.51·31-s + 1.87·41-s − 2/7·49-s − 1.04·59-s + 3.32·61-s − 1.42·71-s − 2.02·79-s + 1.27·89-s + 2.38·101-s − 3.25·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.356333544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.356333544\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 123 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.570934080550980413168953813353, −7.79593909787110178279249813167, −7.64503171774460451266334485033, −7.34332201287685559925364283471, −6.99970705041457890283100079190, −6.59068482433198771955878496084, −6.13141467073513282250848454377, −5.82533640025509501440969514072, −5.59999648962559276088828868777, −4.97266406245091826938568769950, −4.86075099394627336774032390671, −4.11691085566333923827286839761, −3.93053310083182628517750639645, −3.57829159514621225164626331239, −3.05355969380010303752950436859, −2.63152661745988196379328072350, −2.19025728196716371850517195257, −1.31353487338724060151061674662, −1.28528210237523047739542514516, −0.52823090543578266186422143339,
0.52823090543578266186422143339, 1.28528210237523047739542514516, 1.31353487338724060151061674662, 2.19025728196716371850517195257, 2.63152661745988196379328072350, 3.05355969380010303752950436859, 3.57829159514621225164626331239, 3.93053310083182628517750639645, 4.11691085566333923827286839761, 4.86075099394627336774032390671, 4.97266406245091826938568769950, 5.59999648962559276088828868777, 5.82533640025509501440969514072, 6.13141467073513282250848454377, 6.59068482433198771955878496084, 6.99970705041457890283100079190, 7.34332201287685559925364283471, 7.64503171774460451266334485033, 7.79593909787110178279249813167, 8.570934080550980413168953813353
