| L(s) = 1 | + 46·3-s − 150·5-s + 494·7-s + 1.05e3·9-s − 2.80e3·11-s − 5.40e3·13-s − 6.90e3·15-s + 5.18e3·17-s + 2.27e4·21-s − 4.27e3·23-s + 6.87e3·25-s + 3.35e4·27-s + 7.56e4·31-s − 1.28e5·33-s − 7.41e4·35-s + 7.42e4·37-s − 2.48e5·39-s − 7.08e4·41-s − 7.83e4·43-s − 1.58e5·45-s − 1.90e5·47-s + 1.22e5·49-s + 2.38e5·51-s + 7.20e4·53-s + 4.20e5·55-s + 1.66e5·61-s + 5.22e5·63-s + ⋯ |
| L(s) = 1 | + 1.70·3-s − 6/5·5-s + 1.44·7-s + 1.45·9-s − 2.10·11-s − 2.46·13-s − 2.04·15-s + 1.05·17-s + 2.45·21-s − 0.351·23-s + 0.439·25-s + 1.70·27-s + 2.54·31-s − 3.58·33-s − 1.72·35-s + 1.46·37-s − 4.19·39-s − 1.02·41-s − 0.985·43-s − 1.74·45-s − 1.83·47-s + 1.03·49-s + 1.79·51-s + 0.483·53-s + 2.52·55-s + 0.734·61-s + 2.09·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.762876738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.762876738\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 6 p^{2} T + p^{6} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 46 T + 1058 T^{2} - 46 p^{6} T^{3} + p^{12} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 494 T + 122018 T^{2} - 494 p^{6} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 1402 T + p^{6} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 5406 T + 14612418 T^{2} + 5406 p^{6} T^{3} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5186 T + 13447298 T^{2} - 5186 p^{6} T^{3} + p^{12} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 91133362 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4274 T + 9133538 T^{2} + 4274 p^{6} T^{3} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 258176242 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 37838 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 74226 T + 2754749538 T^{2} - 74226 p^{6} T^{3} + p^{12} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 35438 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 78354 T + 3069674658 T^{2} + 78354 p^{6} T^{3} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 190386 T + 18123414498 T^{2} + 190386 p^{6} T^{3} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 72034 T + 2594448578 T^{2} - 72034 p^{6} T^{3} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 83067945682 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 83322 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 121666 T + 7401307778 T^{2} + 121666 p^{6} T^{3} + p^{12} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 40318 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 258046 T + 33293869058 T^{2} + 258046 p^{6} T^{3} + p^{12} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 210927781442 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 228846 T + 26185245858 T^{2} - 228846 p^{6} T^{3} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 958888783522 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1065666 T + 567822011778 T^{2} - 1065666 p^{6} T^{3} + p^{12} T^{4} \) |
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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
−13.56387913381748023503279333570, −13.00345734186962658139258175242, −12.32272252864706747799777435692, −11.86472308183064724910036626088, −11.48118797350756122903326389227, −10.41892693275624650907976392529, −10.09893461603040283518573104103, −9.648894560115212198907832042263, −8.515116551533585483112524831943, −8.138952864151834356928186734411, −7.76143273259698223415655886032, −7.73842305188523514648481902228, −6.78737641092737816603556904857, −5.18249967940827970999415331610, −4.89695970585162317341108092755, −4.27772543215130987762098292790, −2.93205834552955466235735102161, −2.82726298019709698696922407505, −1.94265533622230572911474186215, −0.53895663506303471260929977951,
0.53895663506303471260929977951, 1.94265533622230572911474186215, 2.82726298019709698696922407505, 2.93205834552955466235735102161, 4.27772543215130987762098292790, 4.89695970585162317341108092755, 5.18249967940827970999415331610, 6.78737641092737816603556904857, 7.73842305188523514648481902228, 7.76143273259698223415655886032, 8.138952864151834356928186734411, 8.515116551533585483112524831943, 9.648894560115212198907832042263, 10.09893461603040283518573104103, 10.41892693275624650907976392529, 11.48118797350756122903326389227, 11.86472308183064724910036626088, 12.32272252864706747799777435692, 13.00345734186962658139258175242, 13.56387913381748023503279333570
