| L(s) = 1 | + 76·5-s + 98·7-s + 564·11-s + 516·13-s − 76·17-s + 2.36e3·19-s − 2.03e3·23-s + 1.89e3·25-s + 8.32e3·29-s − 6.28e3·31-s + 7.44e3·35-s − 2.46e3·37-s − 1.43e4·41-s + 2.31e4·43-s + 1.27e4·47-s + 7.20e3·49-s − 9.89e3·53-s + 4.28e4·55-s + 6.08e4·59-s + 1.61e4·61-s + 3.92e4·65-s − 6.25e4·67-s − 732·71-s + 1.24e3·73-s + 5.52e4·77-s + 1.16e4·79-s + 1.32e5·83-s + ⋯ |
| L(s) = 1 | + 1.35·5-s + 0.755·7-s + 1.40·11-s + 0.846·13-s − 0.0637·17-s + 1.49·19-s − 0.802·23-s + 0.607·25-s + 1.83·29-s − 1.17·31-s + 1.02·35-s − 0.295·37-s − 1.32·41-s + 1.90·43-s + 0.839·47-s + 3/7·49-s − 0.483·53-s + 1.91·55-s + 2.27·59-s + 0.556·61-s + 1.15·65-s − 1.70·67-s − 0.0172·71-s + 0.0273·73-s + 1.06·77-s + 0.209·79-s + 2.10·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(8.509236791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.509236791\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 76 T + 3878 T^{2} - 76 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 564 T + 306226 T^{2} - 564 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 516 T + 748094 T^{2} - 516 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 76 T + 2532062 T^{2} + 76 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2360 T + 6283542 T^{2} - 2360 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2036 T + 7494314 T^{2} + 2036 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8320 T + 56801498 T^{2} - 8320 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6280 T + 40192206 T^{2} + 6280 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2460 T + 126463214 T^{2} + 2460 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14308 T + 112999982 T^{2} + 14308 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 23128 T + 423835398 T^{2} - 23128 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12712 T + 311021006 T^{2} - 12712 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9896 T + 858294074 T^{2} + 9896 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1032 p T + 2207083270 T^{2} - 1032 p^{6} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16172 T - 115264002 T^{2} - 16172 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 62568 T + 3672833270 T^{2} + 62568 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 732 T + 3302973034 T^{2} + 732 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1244 T + 3694959894 T^{2} - 1244 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11600 T - 2536172706 T^{2} - 11600 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 132288 T + 12228687622 T^{2} - 132288 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 93452 T + 11465536430 T^{2} - 93452 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 272932 T + 35426184966 T^{2} + 272932 p^{5} T^{3} + p^{10} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.09643303809342031703604252743, −10.09501710988274785627117458103, −9.305822788584994714626157069616, −9.185276349563797405595232866432, −8.567394821905978155634652467762, −8.340712896067558395993807114427, −7.47391450322561201102782909923, −7.26425617810009660991305948933, −6.43259419266421414122982247652, −6.32936164210507515187339512786, −5.53695850994503898476959626271, −5.49373491381535862314628613065, −4.67659729049677365786327388397, −4.15466131596546652783247191916, −3.56181410134512668492532331681, −3.00941605908847930192460466274, −2.01298709981737575931219261660, −1.89170276090216357922333399502, −1.01118824270962119574477749212, −0.816362630894719059567121498376,
0.816362630894719059567121498376, 1.01118824270962119574477749212, 1.89170276090216357922333399502, 2.01298709981737575931219261660, 3.00941605908847930192460466274, 3.56181410134512668492532331681, 4.15466131596546652783247191916, 4.67659729049677365786327388397, 5.49373491381535862314628613065, 5.53695850994503898476959626271, 6.32936164210507515187339512786, 6.43259419266421414122982247652, 7.26425617810009660991305948933, 7.47391450322561201102782909923, 8.340712896067558395993807114427, 8.567394821905978155634652467762, 9.185276349563797405595232866432, 9.305822788584994714626157069616, 10.09501710988274785627117458103, 10.09643303809342031703604252743
