| L(s) = 1 | − 12·3-s + 90·9-s − 48·11-s + 120·17-s − 48·19-s − 164·25-s − 540·27-s + 576·33-s + 408·41-s − 528·43-s − 268·49-s − 1.44e3·51-s + 576·57-s − 1.87e3·59-s − 2.35e3·67-s + 968·73-s + 1.96e3·75-s + 2.83e3·81-s − 3.40e3·83-s − 3.67e3·89-s − 1.48e3·97-s − 4.32e3·99-s − 2.83e3·107-s − 3.96e3·113-s − 2.86e3·121-s − 4.89e3·123-s + 127-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 10/3·9-s − 1.31·11-s + 1.71·17-s − 0.579·19-s − 1.31·25-s − 3.84·27-s + 3.03·33-s + 1.55·41-s − 1.87·43-s − 0.781·49-s − 3.95·51-s + 1.33·57-s − 4.13·59-s − 4.28·67-s + 1.55·73-s + 3.02·75-s + 35/9·81-s − 4.50·83-s − 4.37·89-s − 1.54·97-s − 4.38·99-s − 2.55·107-s − 3.29·113-s − 2.14·121-s − 3.58·123-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 + 164 T^{2} + 19542 T^{4} + 164 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 268 T^{2} - 41658 T^{4} + 268 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 24 T + 2294 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 148 T^{2} + 3687126 T^{4} + 148 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 60 T + 6118 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 24 T + 9254 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 4900 T^{2} + 206522790 T^{4} - 4900 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 56516 T^{2} + 1986668214 T^{4} + 56516 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 63916 T^{2} + 2595558 p^{2} T^{4} + 63916 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 95476 T^{2} + 7028163510 T^{4} + 95476 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 204 T + 806 p T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 264 T + 171830 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 112892 T^{2} + 23215757766 T^{4} + 112892 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 507620 T^{2} + 107623720470 T^{4} + 507620 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 936 T + 498710 T^{2} + 936 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 132332 T^{2} + 107394800406 T^{4} - 132332 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 1176 T + 873542 T^{2} + 1176 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 4612 p T^{2} + 275267100390 T^{4} + 4612 p^{7} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 484 T + 541686 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 932332 T^{2} + 435081520806 T^{4} + 932332 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 1704 T + 1721510 T^{2} + 1704 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 1836 T + 2086774 T^{2} + 1836 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 740 T + 1943814 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.61220000136368121231905024533, −7.14886467714468270679287562193, −6.87456356658864001784619793767, −6.72029208266370278526931025999, −6.54577222628462651265893047953, −6.09171424837873078478019261197, −6.05128342612626067975836643191, −5.82242912065902434976554895984, −5.59825716626785786740901378180, −5.42617914250620112174331567877, −5.15758555531918651461485660041, −5.04855618871435922608152888788, −4.70065704031908553793615787680, −4.27392898186748340301400836707, −4.15057413554570771578051467932, −4.14499911445470323226834396269, −3.68477075505199428458170502344, −3.05719960040914241416425080409, −3.03066819772129874927751003009, −2.69573101912674409562873122529, −2.48836795981775393367591526970, −1.67229704525362929183338713809, −1.43873470305663176195715112540, −1.33446789070603182344933788287, −1.20077684798584418431450561337, 0, 0, 0, 0,
1.20077684798584418431450561337, 1.33446789070603182344933788287, 1.43873470305663176195715112540, 1.67229704525362929183338713809, 2.48836795981775393367591526970, 2.69573101912674409562873122529, 3.03066819772129874927751003009, 3.05719960040914241416425080409, 3.68477075505199428458170502344, 4.14499911445470323226834396269, 4.15057413554570771578051467932, 4.27392898186748340301400836707, 4.70065704031908553793615787680, 5.04855618871435922608152888788, 5.15758555531918651461485660041, 5.42617914250620112174331567877, 5.59825716626785786740901378180, 5.82242912065902434976554895984, 6.05128342612626067975836643191, 6.09171424837873078478019261197, 6.54577222628462651265893047953, 6.72029208266370278526931025999, 6.87456356658864001784619793767, 7.14886467714468270679287562193, 7.61220000136368121231905024533
