| L(s) = 1 | − 67·3-s + 4.09e3·4-s + 4.07e4·5-s + 1.77e5·9-s − 2.74e5·12-s − 2.73e6·15-s + 4.19e6·16-s + 1.67e8·20-s + 8.75e8·25-s − 3.53e7·27-s + 2.92e8·31-s + 7.25e8·36-s − 7.50e8·37-s + 7.22e9·45-s + 7.03e9·47-s − 2.81e8·48-s − 3.95e9·49-s − 2.41e10·59-s − 1.11e10·60-s − 1.71e10·64-s − 1.36e10·67-s − 5.86e10·75-s + 1.71e11·80-s + 2.36e9·81-s − 1.95e10·93-s + 1.51e11·97-s + 3.58e12·100-s + ⋯ |
| L(s) = 1 | − 0.159·3-s + 2·4-s + 5.83·5-s + 9-s − 0.318·12-s − 0.928·15-s + 16-s + 11.6·20-s + 17.9·25-s − 0.473·27-s + 1.83·31-s + 2·36-s − 1.77·37-s + 5.83·45-s + 4.47·47-s − 0.159·48-s − 2·49-s − 4.39·59-s − 1.85·60-s − 2·64-s − 1.23·67-s − 2.85·75-s + 5.83·80-s + 0.0753·81-s − 0.292·93-s + 1.79·97-s + 35.8·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(48.54022860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(48.54022860\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_2^2$ | \( 1 + 67 T - 172658 T^{2} + 67 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 13593 T + p^{11} T^{2} )^{2}( 1 - 13593 T + 135941524 T^{2} - 13593 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 45497817 T + 1117241593851562 T^{2} - 45497817 p^{11} T^{3} + p^{22} T^{4} )( 1 + 45497817 T + 1117241593851562 T^{2} + 45497817 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 - p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 292394195 T + p^{11} T^{2} )^{2}( 1 + 292394195 T + 60085888373293194 T^{2} + 292394195 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 + 375115349 T - 37206096724068612 T^{2} + 375115349 p^{11} T^{3} + p^{22} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2346355716 T + p^{11} T^{2} )^{2}( 1 - 2346355716 T + 3033225930921860353 T^{2} - 2346355716 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6084080382 T + 27746998165265074327 T^{2} - 6084080382 p^{11} T^{3} + p^{22} T^{4} )( 1 + 6084080382 T + 27746998165265074327 T^{2} + 6084080382 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 8051651835 T + p^{11} T^{2} )^{2}( 1 + 8051651835 T + 34673208827321024566 T^{2} + 8051651835 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 13691120599 T + p^{11} T^{2} )^{2}( 1 - 13691120599 T + 65316650351394101718 T^{2} - 13691120599 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 71 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 28030204947 T + \)\(55\!\cdots\!38\)\( T^{2} - 28030204947 p^{11} T^{3} + p^{22} T^{4} )( 1 + 28030204947 T + \)\(55\!\cdots\!38\)\( T^{2} + 28030204947 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 73 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 76570141041 T + p^{11} T^{2} )^{2}( 1 + 76570141041 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 151692012401 T + p^{11} T^{2} )^{2}( 1 + 151692012401 T + \)\(15\!\cdots\!48\)\( T^{2} + 151692012401 p^{11} T^{3} + p^{22} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66671279127340539591823135272, −7.61898714230456670574193277720, −7.11952984698771850825280034682, −7.10947060367065335419891305194, −6.47443003106196028259880185688, −6.28258199871014437456085904625, −6.20336596262346050023480067644, −6.17811747305470507831095745911, −5.91282122432393852203941894400, −5.48262551801367838678806068810, −5.16518711079325029464937302719, −4.87844519184767006614584388676, −4.66721550610601309857764080450, −4.16931563868504664878536886501, −3.29526341663225989448393599385, −3.07632713999242314844501948897, −2.60086839290420812169942906711, −2.44208822734320687711108917659, −2.27201298788849660013012326783, −1.96986766852750292800608993173, −1.77118125092562715675569449877, −1.37291172447481103640456124957, −1.23478950609980843215616785471, −1.20499611443342676478856522025, −0.33541368695806179145340345446,
0.33541368695806179145340345446, 1.20499611443342676478856522025, 1.23478950609980843215616785471, 1.37291172447481103640456124957, 1.77118125092562715675569449877, 1.96986766852750292800608993173, 2.27201298788849660013012326783, 2.44208822734320687711108917659, 2.60086839290420812169942906711, 3.07632713999242314844501948897, 3.29526341663225989448393599385, 4.16931563868504664878536886501, 4.66721550610601309857764080450, 4.87844519184767006614584388676, 5.16518711079325029464937302719, 5.48262551801367838678806068810, 5.91282122432393852203941894400, 6.17811747305470507831095745911, 6.20336596262346050023480067644, 6.28258199871014437456085904625, 6.47443003106196028259880185688, 7.10947060367065335419891305194, 7.11952984698771850825280034682, 7.61898714230456670574193277720, 7.66671279127340539591823135272
