| L(s) = 1 | + 2·3-s − 9-s − 4·11-s − 4·13-s + 4·17-s + 14·23-s − 6·27-s − 10·29-s − 4·31-s − 8·33-s − 8·39-s + 6·41-s − 6·43-s − 12·47-s + 8·51-s − 8·59-s + 2·61-s + 6·67-s + 28·69-s − 24·71-s + 4·73-s − 8·79-s − 4·81-s + 18·83-s − 20·87-s − 22·89-s − 8·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.970·17-s + 2.91·23-s − 1.15·27-s − 1.85·29-s − 0.718·31-s − 1.39·33-s − 1.28·39-s + 0.937·41-s − 0.914·43-s − 1.75·47-s + 1.12·51-s − 1.04·59-s + 0.256·61-s + 0.733·67-s + 3.37·69-s − 2.84·71-s + 0.468·73-s − 0.900·79-s − 4/9·81-s + 1.97·83-s − 2.14·87-s − 2.33·89-s − 0.829·93-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 14 T + 93 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T - 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 91 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 229 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 22 T + 3 p T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−7.36704910371947464701627265874, −7.31190273626626586122409322862, −7.07526705545864904323204606151, −6.67355071855595058923641126825, −5.91118536937455914598724625034, −5.78023028878370388026184991397, −5.45970871883817431002797705811, −5.07798928232015481359769948541, −4.64586212396625846765261830298, −4.63910898773724583214304905114, −3.72359487845364375365054891267, −3.46894692130880219024768769752, −3.05556635382773581164498866073, −2.98188273920396512757567966598, −2.41389015489771500166816105683, −2.22158897405785341119485150975, −1.48617882024117632317609111240, −1.13761349690508980669908107473, 0, 0,
1.13761349690508980669908107473, 1.48617882024117632317609111240, 2.22158897405785341119485150975, 2.41389015489771500166816105683, 2.98188273920396512757567966598, 3.05556635382773581164498866073, 3.46894692130880219024768769752, 3.72359487845364375365054891267, 4.63910898773724583214304905114, 4.64586212396625846765261830298, 5.07798928232015481359769948541, 5.45970871883817431002797705811, 5.78023028878370388026184991397, 5.91118536937455914598724625034, 6.67355071855595058923641126825, 7.07526705545864904323204606151, 7.31190273626626586122409322862, 7.36704910371947464701627265874
