| L(s) = 1 | + 4·2-s + 9·3-s + 12·4-s − 9·5-s + 36·6-s − 18·7-s + 32·8-s + 25·9-s − 36·10-s − 17·11-s + 108·12-s + 17·13-s − 72·14-s − 81·15-s + 80·16-s − 80·17-s + 100·18-s + 38·19-s − 108·20-s − 162·21-s − 68·22-s + 73·23-s + 288·24-s − 25·25-s + 68·26-s − 36·27-s − 216·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 3/2·4-s − 0.804·5-s + 2.44·6-s − 0.971·7-s + 1.41·8-s + 0.925·9-s − 1.13·10-s − 0.465·11-s + 2.59·12-s + 0.362·13-s − 1.37·14-s − 1.39·15-s + 5/4·16-s − 1.14·17-s + 1.30·18-s + 0.458·19-s − 1.20·20-s − 1.68·21-s − 0.658·22-s + 0.661·23-s + 2.44·24-s − 1/5·25-s + 0.512·26-s − 0.256·27-s − 1.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.303528832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.303528832\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - p^{2} T + 56 T^{2} - p^{5} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 9 T + 106 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 18 T + 475 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 17 T + 2716 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 17 T + 1382 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 80 T + 11353 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 73 T + 22582 T^{2} - 73 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 40732 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 212 T + 35486 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 192 T + 96214 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 50 T + 136642 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 677 T + 272702 T^{2} - 677 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 389 T + 153478 T^{2} + 389 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 23 p T + 663970 T^{2} + 23 p^{4} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 287 T + 419944 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 313 T + 253596 T^{2} - 313 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1223 T + 867254 T^{2} - 1223 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 200 T + 480250 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 378 T + 195883 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1350 T + 1380310 T^{2} - 1350 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 670 T + 568942 T^{2} + 670 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 236 T + 778834 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1294 T + 2054082 T^{2} - 1294 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−15.72995924244702178401287918599, −15.58716924581823310625876813870, −14.71090850537627076555606662756, −14.38500981277940223423798464016, −13.68656564713630820883903045839, −13.38775329037008038079963478557, −12.72576187683773906907546760818, −12.30313414350694990763024205308, −11.12894287555751293338822293772, −11.08453363412813674368454222799, −9.697880035208362787704356901554, −9.261357644577212813933289374130, −8.084959882271028323143027320107, −7.996750157870826228222825275427, −6.83945744702810489928646858932, −6.20969467878938420561383725721, −4.89012081848617119485394458011, −3.93149329831377357804181671820, −3.16303426781052515390090952770, −2.55507133412447475098459378132,
2.55507133412447475098459378132, 3.16303426781052515390090952770, 3.93149329831377357804181671820, 4.89012081848617119485394458011, 6.20969467878938420561383725721, 6.83945744702810489928646858932, 7.996750157870826228222825275427, 8.084959882271028323143027320107, 9.261357644577212813933289374130, 9.697880035208362787704356901554, 11.08453363412813674368454222799, 11.12894287555751293338822293772, 12.30313414350694990763024205308, 12.72576187683773906907546760818, 13.38775329037008038079963478557, 13.68656564713630820883903045839, 14.38500981277940223423798464016, 14.71090850537627076555606662756, 15.58716924581823310625876813870, 15.72995924244702178401287918599
