| L(s) = 1 | + 64·2-s + 482·3-s + 1.02e3·4-s + 3.08e4·6-s − 6.55e4·8-s + 5.90e4·9-s − 9.74e4·11-s + 4.93e5·12-s − 4.19e6·16-s − 1.64e6·17-s + 3.77e6·18-s − 6.70e6·19-s − 6.23e6·22-s − 3.15e7·24-s − 1.95e7·25-s − 2.65e7·27-s − 6.71e7·32-s − 4.69e7·33-s − 1.05e8·34-s + 6.04e7·36-s − 4.29e8·38-s − 3.77e7·41-s + 2.14e8·43-s − 9.97e7·44-s − 2.02e9·48-s − 5.64e8·49-s − 1.25e9·50-s + ⋯ |
| L(s) = 1 | + 2·2-s + 1.98·3-s + 4-s + 3.96·6-s − 2·8-s + 9-s − 0.604·11-s + 1.98·12-s − 4·16-s − 1.16·17-s + 2·18-s − 2.70·19-s − 1.20·22-s − 3.96·24-s − 2·25-s − 1.85·27-s − 2·32-s − 1.19·33-s − 2.32·34-s + 36-s − 5.41·38-s − 0.326·41-s + 1.45·43-s − 0.604·44-s − 7.93·48-s − 2·49-s − 4·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.02083313419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02083313419\) |
| \(L(6)\) |
|
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 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 482 T + 173275 T^{2} - 482 p^{10} T^{3} + p^{20} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )^{2}( 1 + p^{5} T + p^{10} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )^{2}( 1 + p^{5} T + p^{10} T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 97426 T + p^{10} T^{2} )^{2}( 1 - 97426 T - 16445599125 T^{2} - 97426 p^{10} T^{3} + p^{20} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )^{2}( 1 + p^{5} T + p^{10} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 823682 T - 1337541863325 T^{2} + 823682 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 3353726 T + 5116411825275 T^{2} + 3353726 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )^{2}( 1 + p^{5} T + p^{10} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )^{2}( 1 + p^{5} T + p^{10} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )^{2}( 1 + p^{5} T + p^{10} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{4}( 1 + p^{5} T )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 37778926 T + p^{10} T^{2} )^{2}( 1 - 37778926 T - 11995412060438925 T^{2} - 37778926 p^{10} T^{3} + p^{20} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 214485614 T + p^{10} T^{2} )^{2}( 1 + 214485614 T + 24392596299672747 T^{2} + 214485614 p^{10} T^{3} + p^{20} T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )^{2}( 1 + p^{5} T + p^{10} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{4}( 1 + p^{5} T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 921043598 T + p^{10} T^{2} )^{2}( 1 + 921043598 T + 337204556116144203 T^{2} + 921043598 p^{10} T^{3} + p^{20} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )^{2}( 1 + p^{5} T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 1813708382 T + p^{10} T^{2} )^{2}( 1 + 1813708382 T + 1466700290385296475 T^{2} + 1813708382 p^{10} T^{3} + p^{20} T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{4}( 1 + p^{5} T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 1605781582 T - 1719091340613134925 T^{2} - 1605781582 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )^{2}( 1 + p^{5} T + p^{10} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 96051518 T - 15506815293095749125 T^{2} + 96051518 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 11116019374 T + p^{10} T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 9872978014 T + p^{10} T^{2} )^{2}( 1 - 9872978014 T + 23733282175434558147 T^{2} - 9872978014 p^{10} T^{3} + p^{20} T^{4} ) \) |
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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
−8.523873470314555454281737268008, −8.446651239314951896441015539193, −8.202880258447012692421269444853, −8.063716449677945250042952378200, −7.54071284459322084181487499333, −6.91781258565010649293541603030, −6.80249584631180655716627022240, −6.25687778497660138935030319491, −6.19239051390256829571686086906, −5.70312021377649958154446021804, −5.38426326519660272817839861223, −5.11720407540557915777601175316, −4.73609063184352527249551239612, −4.04569422565809740184132164505, −3.94496866437971251355083354710, −3.91353548660144004501496057834, −3.79212976375337459534655000506, −2.75647676127820277002644595525, −2.72691651574352998093534322105, −2.48407280373415796793318411258, −2.45359875581041379202342790973, −1.62357065231206202465965146288, −1.48192654459128824804106218052, −0.27881876330012492393491559282, −0.02129365013720572231883783604,
0.02129365013720572231883783604, 0.27881876330012492393491559282, 1.48192654459128824804106218052, 1.62357065231206202465965146288, 2.45359875581041379202342790973, 2.48407280373415796793318411258, 2.72691651574352998093534322105, 2.75647676127820277002644595525, 3.79212976375337459534655000506, 3.91353548660144004501496057834, 3.94496866437971251355083354710, 4.04569422565809740184132164505, 4.73609063184352527249551239612, 5.11720407540557915777601175316, 5.38426326519660272817839861223, 5.70312021377649958154446021804, 6.19239051390256829571686086906, 6.25687778497660138935030319491, 6.80249584631180655716627022240, 6.91781258565010649293541603030, 7.54071284459322084181487499333, 8.063716449677945250042952378200, 8.202880258447012692421269444853, 8.446651239314951896441015539193, 8.523873470314555454281737268008
