| L(s) = 1 | + 3·5-s + 51·11-s + 61·13-s − 24·17-s + 169·19-s + 192·23-s − 115·25-s + 39·29-s − 92·31-s − 173·37-s − 174·41-s − 497·43-s − 180·47-s − 285·53-s + 153·55-s + 1.26e3·59-s + 328·61-s + 183·65-s − 875·67-s + 1.40e3·71-s − 1.36e3·73-s − 182·79-s + 399·83-s − 72·85-s − 822·89-s + 507·95-s + 841·97-s + ⋯ |
| L(s) = 1 | + 0.268·5-s + 1.39·11-s + 1.30·13-s − 0.342·17-s + 2.04·19-s + 1.74·23-s − 0.919·25-s + 0.249·29-s − 0.533·31-s − 0.768·37-s − 0.662·41-s − 1.76·43-s − 0.558·47-s − 0.738·53-s + 0.375·55-s + 2.80·59-s + 0.688·61-s + 0.349·65-s − 1.59·67-s + 2.34·71-s − 2.18·73-s − 0.259·79-s + 0.527·83-s − 0.0918·85-s − 0.979·89-s + 0.547·95-s + 0.880·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.583486715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.583486715\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 3 T + 124 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 51 T + 3184 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 61 T + 2118 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 24 T + 1762 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 169 T + 20730 T^{2} - 169 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 39 T + 12094 T^{2} - 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 92 T + 48873 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 173 T + 62490 T^{2} + 173 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 174 T - 2846 T^{2} + 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 497 T + 174468 T^{2} + 497 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 180 T + 182914 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 285 T + 224566 T^{2} + 285 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1269 T + 13766 p T^{2} - 1269 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 328 T + 314646 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 875 T + 782544 T^{2} + 875 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1404 T + 1077298 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1361 T + 1184556 T^{2} + 1361 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 182 T + 992307 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 399 T + 946240 T^{2} - 399 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 822 T + 1516786 T^{2} + 822 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 841 T + 1945608 T^{2} - 841 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.043870413797742506119105227184, −8.843936182139608526676785779832, −8.406197711863902879593614381215, −8.112555006259609676034533568755, −7.33168268264071741847018725930, −7.16632895284681681750235456354, −6.80119878451450598183650776440, −6.34119735755441186842475952228, −5.96631575132704157135759826303, −5.51373486549161232979806143276, −4.93410294374654109307218445959, −4.87818999672405530096632909682, −3.90999532815131342013252620750, −3.71364937799688449384097425455, −3.23991109019821957333915354255, −2.91740681688651484027174819094, −1.79368285463605973708956418384, −1.71733159692604661448146171938, −0.985660837090024875727802129952, −0.59386709720937646950305169960,
0.59386709720937646950305169960, 0.985660837090024875727802129952, 1.71733159692604661448146171938, 1.79368285463605973708956418384, 2.91740681688651484027174819094, 3.23991109019821957333915354255, 3.71364937799688449384097425455, 3.90999532815131342013252620750, 4.87818999672405530096632909682, 4.93410294374654109307218445959, 5.51373486549161232979806143276, 5.96631575132704157135759826303, 6.34119735755441186842475952228, 6.80119878451450598183650776440, 7.16632895284681681750235456354, 7.33168268264071741847018725930, 8.112555006259609676034533568755, 8.406197711863902879593614381215, 8.843936182139608526676785779832, 9.043870413797742506119105227184
