| L(s) = 1 | − 12·3-s + 50·5-s + 52·7-s + 138·9-s + 560·11-s + 1.38e3·13-s − 600·15-s + 148·17-s − 1.00e3·19-s − 624·21-s − 2.45e3·23-s + 1.87e3·25-s − 4.50e3·27-s + 1.34e3·29-s − 2.24e3·31-s − 6.72e3·33-s + 2.60e3·35-s − 5.94e3·37-s − 1.66e4·39-s + 2.30e4·41-s + 1.76e4·43-s + 6.90e3·45-s − 2.90e3·47-s − 2.69e4·49-s − 1.77e3·51-s − 5.41e3·53-s + 2.80e4·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 0.894·5-s + 0.401·7-s + 0.567·9-s + 1.39·11-s + 2.27·13-s − 0.688·15-s + 0.124·17-s − 0.635·19-s − 0.308·21-s − 0.966·23-s + 3/5·25-s − 1.18·27-s + 0.295·29-s − 0.420·31-s − 1.07·33-s + 0.358·35-s − 0.713·37-s − 1.75·39-s + 2.14·41-s + 1.45·43-s + 0.507·45-s − 0.192·47-s − 1.60·49-s − 0.0956·51-s − 0.264·53-s + 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.408631218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.408631218\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 4 p T + 2 p T^{2} + 4 p^{6} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 52 T + 29646 T^{2} - 52 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 560 T + 381926 T^{2} - 560 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1388 T + 1149918 T^{2} - 1388 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 148 T - 795706 T^{2} - 148 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1000 T + 4533462 T^{2} + 1000 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2452 T + 6569198 T^{2} + 2452 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 1340 T - 8758306 T^{2} - 1340 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2248 T + 57017022 T^{2} + 2248 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5940 T + 123434318 T^{2} + 5940 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 23076 T + 352280470 T^{2} - 23076 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 17684 T + 312898614 T^{2} - 17684 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2908 T + 56660030 T^{2} + 2908 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5412 T + 693247822 T^{2} + 5412 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 62584 T + 2277965606 T^{2} - 62584 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14108 T + 1110042462 T^{2} - 14108 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 85412 T + 4371910566 T^{2} + 85412 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 47208 T + 4011779662 T^{2} - 47208 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 924 p T + 4400780438 T^{2} + 924 p^{6} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 65904 T + 3994274078 T^{2} + 65904 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 108724 T + 10572459494 T^{2} - 108724 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 55020 T + 10818978262 T^{2} + 55020 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 147668 T + 11612429670 T^{2} - 147668 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−15.62375432497308448995676140324, −14.63596785280524036363235531400, −14.30480817048809154064998370715, −13.67907441729859442256179111506, −13.02789333016761728421589039008, −12.57677095638850143305280077938, −11.52366841667499636215992174181, −11.38187244956132201060069553240, −10.63299398244359297175836293002, −10.01589605809223993077363237156, −9.119816626608655154943870402951, −8.741664202543982978117411245315, −7.75242387354525337515079351312, −6.69028411269434210771731885993, −6.02982442502564268997390633906, −5.77491841906999518906764568867, −4.38756066350892397807912794873, −3.70113498533187696805245675049, −1.85823215542219965787158669896, −1.03926941274387637888507780149,
1.03926941274387637888507780149, 1.85823215542219965787158669896, 3.70113498533187696805245675049, 4.38756066350892397807912794873, 5.77491841906999518906764568867, 6.02982442502564268997390633906, 6.69028411269434210771731885993, 7.75242387354525337515079351312, 8.741664202543982978117411245315, 9.119816626608655154943870402951, 10.01589605809223993077363237156, 10.63299398244359297175836293002, 11.38187244956132201060069553240, 11.52366841667499636215992174181, 12.57677095638850143305280077938, 13.02789333016761728421589039008, 13.67907441729859442256179111506, 14.30480817048809154064998370715, 14.63596785280524036363235531400, 15.62375432497308448995676140324
