| L(s) = 1 | + 1.01e3·13-s − 3.12e4·25-s + 1.78e5·37-s − 1.53e5·49-s + 8.41e5·61-s − 1.27e6·73-s + 1.12e5·97-s − 4.34e6·109-s + 3.54e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8.88e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 0.460·13-s − 2·25-s + 3.52·37-s − 1.30·49-s + 3.70·61-s − 3.28·73-s + 0.123·97-s − 3.35·109-s + 2·121-s − 1.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.561030880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.561030880\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 286 T + p^{6} T^{2} )( 1 + 286 T + p^{6} T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 506 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 10582 T + p^{6} T^{2} )( 1 + 10582 T + p^{6} T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 35282 T + p^{6} T^{2} )( 1 + 35282 T + p^{6} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 89206 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 111386 T + p^{6} T^{2} )( 1 + 111386 T + p^{6} T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 420838 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 172874 T + p^{6} T^{2} )( 1 + 172874 T + p^{6} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 638066 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 204622 T + p^{6} T^{2} )( 1 + 204622 T + p^{6} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 56446 T + p^{6} T^{2} )^{2} \) |
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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
−10.04702070512571584480784737319, −9.536871187646125319669443002477, −9.289627316631763258219267608269, −8.471200670557012517643166645796, −8.314246424656962931025877455605, −7.76263471056334023878151358283, −7.43768934549868427723407253652, −6.82764369760900709602906767231, −6.31368565358844158059062503892, −5.78836579954562136880957347737, −5.68782599641704214013825816543, −4.80060606504751717040134010282, −4.37432617851005642414757765254, −3.86004312670936517996057821616, −3.46038067847563556563468064800, −2.47793037816005429681584765641, −2.44419044450745920653523501124, −1.45646112958041163913903466984, −1.05323974040172769267947285848, −0.27169553816373747317597906172,
0.27169553816373747317597906172, 1.05323974040172769267947285848, 1.45646112958041163913903466984, 2.44419044450745920653523501124, 2.47793037816005429681584765641, 3.46038067847563556563468064800, 3.86004312670936517996057821616, 4.37432617851005642414757765254, 4.80060606504751717040134010282, 5.68782599641704214013825816543, 5.78836579954562136880957347737, 6.31368565358844158059062503892, 6.82764369760900709602906767231, 7.43768934549868427723407253652, 7.76263471056334023878151358283, 8.314246424656962931025877455605, 8.471200670557012517643166645796, 9.289627316631763258219267608269, 9.536871187646125319669443002477, 10.04702070512571584480784737319
