| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 2·11-s + 5·16-s + 4·22-s + 4·23-s − 8·25-s + 12·29-s + 6·32-s − 4·37-s + 8·43-s + 6·44-s + 8·46-s − 16·50-s + 24·53-s + 24·58-s + 7·64-s − 4·67-s + 20·71-s − 8·74-s − 16·79-s + 16·86-s + 8·88-s + 12·92-s − 24·100-s + 48·106-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 0.603·11-s + 5/4·16-s + 0.852·22-s + 0.834·23-s − 8/5·25-s + 2.22·29-s + 1.06·32-s − 0.657·37-s + 1.21·43-s + 0.904·44-s + 1.17·46-s − 2.26·50-s + 3.29·53-s + 3.15·58-s + 7/8·64-s − 0.488·67-s + 2.37·71-s − 0.929·74-s − 1.80·79-s + 1.72·86-s + 0.852·88-s + 1.25·92-s − 2.39·100-s + 4.66·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.53233633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.53233633\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
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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
−7.52439934963764878974639506515, −7.39197473456050186700376648207, −7.05320983189886319611053331847, −6.90665375595235005573157433849, −6.22766429434383699223659453906, −6.16945730938830495618789705776, −5.68329620190205009304303433332, −5.63771500837814560750558809974, −4.89129234534473132451797989846, −4.79835299851017484138206132012, −4.35806641354321319274469481185, −4.12104537381210029049728139367, −3.50982926698003481113230121096, −3.44978499480081786200908564789, −2.96492979434940412659366783048, −2.32839567824895550317970906870, −2.22234088658059879164505439615, −1.71120259922571551647729693970, −0.838545002784852569998979181643, −0.791852201091381288167504403472,
0.791852201091381288167504403472, 0.838545002784852569998979181643, 1.71120259922571551647729693970, 2.22234088658059879164505439615, 2.32839567824895550317970906870, 2.96492979434940412659366783048, 3.44978499480081786200908564789, 3.50982926698003481113230121096, 4.12104537381210029049728139367, 4.35806641354321319274469481185, 4.79835299851017484138206132012, 4.89129234534473132451797989846, 5.63771500837814560750558809974, 5.68329620190205009304303433332, 6.16945730938830495618789705776, 6.22766429434383699223659453906, 6.90665375595235005573157433849, 7.05320983189886319611053331847, 7.39197473456050186700376648207, 7.52439934963764878974639506515
