| L(s) = 1 | + 2-s − 6·3-s − 11·4-s − 6·6-s + 4·7-s − 15·8-s + 27·9-s − 22·11-s + 66·12-s + 90·13-s + 4·14-s + 61·16-s + 16·17-s + 27·18-s − 170·19-s − 24·21-s − 22·22-s + 124·23-s + 90·24-s + 90·26-s − 108·27-s − 44·28-s − 158·29-s + 60·31-s + 89·32-s + 132·33-s + 16·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.15·3-s − 1.37·4-s − 0.408·6-s + 0.215·7-s − 0.662·8-s + 9-s − 0.603·11-s + 1.58·12-s + 1.92·13-s + 0.0763·14-s + 0.953·16-s + 0.228·17-s + 0.353·18-s − 2.05·19-s − 0.249·21-s − 0.213·22-s + 1.12·23-s + 0.765·24-s + 0.678·26-s − 0.769·27-s − 0.296·28-s − 1.01·29-s + 0.347·31-s + 0.491·32-s + 0.696·33-s + 0.0807·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.743813736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.743813736\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 622 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 90 T + 6402 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 16 T + 1662 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 170 T + 994 p T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 124 T + 12878 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 158 T + 50106 T^{2} + 158 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 60 T + 59870 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 372 T + 82590 T^{2} - 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 38 T + 37410 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 p T + 160230 T^{2} - 12 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 224 T + 466 p T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 472 T + 190182 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 248 T + 181334 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 72 T - 107850 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 744 T + 738822 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2060 T + 1768494 T^{2} - 2060 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 486 T + 822786 T^{2} - 486 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 642 T + 691166 T^{2} - 642 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 286 T + 392750 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 244 T + 1355190 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 168 T + 1053870 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.876528510536384947953267090958, −9.806986566435316226187738075720, −9.068030042095891331729383022713, −8.954080081240248141494269535549, −8.277786429051531621205711879598, −8.016825592674984079214629781883, −7.65570917597865097032426039793, −6.71788831850485457084068465706, −6.32318897000562416886075608313, −6.23415309639988120884263187498, −5.41503299734325727280676594738, −5.28967833031309285134040649406, −4.51972008784312345191272966679, −4.47846261270584869041862970750, −3.66192660395499297026734709011, −3.53585299880254668214246467297, −2.42307071631622425828895461978, −1.71164066558505707530058870290, −0.66978612872596675055054712958, −0.63797008900176234647170914173,
0.63797008900176234647170914173, 0.66978612872596675055054712958, 1.71164066558505707530058870290, 2.42307071631622425828895461978, 3.53585299880254668214246467297, 3.66192660395499297026734709011, 4.47846261270584869041862970750, 4.51972008784312345191272966679, 5.28967833031309285134040649406, 5.41503299734325727280676594738, 6.23415309639988120884263187498, 6.32318897000562416886075608313, 6.71788831850485457084068465706, 7.65570917597865097032426039793, 8.016825592674984079214629781883, 8.277786429051531621205711879598, 8.954080081240248141494269535549, 9.068030042095891331729383022713, 9.806986566435316226187738075720, 9.876528510536384947953267090958
