| L(s) = 1 | + 3-s − 9-s − 11-s + 13-s + 11·17-s + 6·19-s + 2·23-s + 5·29-s + 4·31-s − 33-s + 39-s − 6·41-s + 6·43-s + 9·47-s + 11·51-s + 18·53-s + 6·57-s + 8·59-s + 22·61-s + 12·67-s + 2·69-s + 8·73-s − 11·79-s − 4·81-s + 24·83-s + 5·87-s − 2·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/3·9-s − 0.301·11-s + 0.277·13-s + 2.66·17-s + 1.37·19-s + 0.417·23-s + 0.928·29-s + 0.718·31-s − 0.174·33-s + 0.160·39-s − 0.937·41-s + 0.914·43-s + 1.31·47-s + 1.54·51-s + 2.47·53-s + 0.794·57-s + 1.04·59-s + 2.81·61-s + 1.46·67-s + 0.240·69-s + 0.936·73-s − 1.23·79-s − 4/9·81-s + 2.63·83-s + 0.536·87-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.829340435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.829340435\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 22 T + 226 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 150 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 212 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−7.86217499834449785748166201244, −7.66594870202169498061069284533, −7.09649512697186944939616124452, −6.91128033461079658207999792441, −6.65418378914650171822743020873, −6.05145586863116930179918523654, −5.67660600547981100308656020804, −5.29175642791039600895899975568, −5.28019561971392628105298986340, −5.01092922542570892458922482269, −4.02638693727193742410500773192, −4.02038187091920110127114256360, −3.59995048207000896712163475080, −3.20649409371169983798220706738, −2.70621060448665808695024167792, −2.62557651961971797966650113412, −2.10032276905432378676392079720, −1.27184810507259228112262381138, −0.861781367386999470685783872783, −0.790127328703066710935470922177,
0.790127328703066710935470922177, 0.861781367386999470685783872783, 1.27184810507259228112262381138, 2.10032276905432378676392079720, 2.62557651961971797966650113412, 2.70621060448665808695024167792, 3.20649409371169983798220706738, 3.59995048207000896712163475080, 4.02038187091920110127114256360, 4.02638693727193742410500773192, 5.01092922542570892458922482269, 5.28019561971392628105298986340, 5.29175642791039600895899975568, 5.67660600547981100308656020804, 6.05145586863116930179918523654, 6.65418378914650171822743020873, 6.91128033461079658207999792441, 7.09649512697186944939616124452, 7.66594870202169498061069284533, 7.86217499834449785748166201244
