| L(s) = 1 | − 234·3-s + 2.06e4·7-s − 4.29e3·9-s − 5.13e5·13-s − 6.39e6·19-s − 4.82e6·21-s + 1.78e7·25-s + 1.48e7·27-s − 4.62e7·31-s + 5.95e7·37-s + 1.20e8·39-s − 4.95e8·43-s − 2.45e8·49-s + 1.49e9·57-s − 2.10e9·61-s − 8.85e7·63-s + 7.22e8·67-s + 7.48e8·73-s − 4.18e9·75-s + 2.27e8·79-s − 3.21e9·81-s − 1.05e10·91-s + 1.08e10·93-s + 5.61e9·97-s − 3.02e10·103-s + 2.79e9·109-s − 1.39e10·111-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 1.22·7-s − 0.0727·9-s − 1.38·13-s − 2.58·19-s − 1.18·21-s + 1.83·25-s + 1.03·27-s − 1.61·31-s + 0.859·37-s + 1.33·39-s − 3.36·43-s − 0.869·49-s + 2.48·57-s − 2.49·61-s − 0.0892·63-s + 0.535·67-s + 0.361·73-s − 1.76·75-s + 0.0740·79-s − 0.922·81-s − 1.69·91-s + 1.55·93-s + 0.654·97-s − 2.60·103-s + 0.181·109-s − 0.827·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.05291516379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05291516379\) |
| \(L(6)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 26 p^{2} T + p^{10} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3579658 p T^{2} + p^{20} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 1474 p T + p^{10} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 33759206158 T^{2} + p^{20} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 256822 T + p^{10} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3721568636738 T^{2} + p^{20} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3196106 T + p^{10} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 13051422193058 T^{2} + p^{20} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 150245316518638 T^{2} + p^{20} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 23140994 T + p^{10} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 29797946 T + p^{10} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 26844419571315362 T^{2} + p^{20} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 247522778 T + p^{10} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3407102420305342 T^{2} + p^{20} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 45853036080911858 T^{2} + p^{20} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 893285092397333042 T^{2} + p^{20} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 1054839766 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 361186198 T + p^{10} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 5642727107803561442 T^{2} + p^{20} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 374437394 T + p^{10} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 113914462 T + p^{10} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6914979814596134738 T^{2} + p^{20} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6513274382958722 p^{2} T^{2} + p^{20} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2809917122 T + p^{10} T^{2} )^{2} \) |
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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
−13.81837654697726439256717260275, −12.80338533522390914811118260705, −12.70110243536718742741446722036, −11.93751236812518996335908342095, −11.38411438703965330651652565785, −10.79019110878862809962500464346, −10.58518188811483143043785292297, −9.679217993912763393461522096933, −8.774392733682256749286396464214, −8.347196557889636507752711957653, −7.61642301911187021857561173963, −6.68283833103627170016699880026, −6.36490538850717369603809054973, −5.14785245548422529167917963689, −4.97547514324320757499531802896, −4.28852633188198016754415243116, −3.01341118428672768628748453439, −2.08264711750180513212237161077, −1.40760360829413777829994532325, −0.07592801262013817262882233575,
0.07592801262013817262882233575, 1.40760360829413777829994532325, 2.08264711750180513212237161077, 3.01341118428672768628748453439, 4.28852633188198016754415243116, 4.97547514324320757499531802896, 5.14785245548422529167917963689, 6.36490538850717369603809054973, 6.68283833103627170016699880026, 7.61642301911187021857561173963, 8.347196557889636507752711957653, 8.774392733682256749286396464214, 9.679217993912763393461522096933, 10.58518188811483143043785292297, 10.79019110878862809962500464346, 11.38411438703965330651652565785, 11.93751236812518996335908342095, 12.70110243536718742741446722036, 12.80338533522390914811118260705, 13.81837654697726439256717260275
