| L(s) = 1 | − 32·2-s + 1.02e3·4-s + 1.18e4·5-s + 1.68e4·7-s − 3.27e4·8-s − 3.80e5·10-s − 7.27e5·11-s − 1.89e6·13-s − 5.37e5·14-s + 1.04e6·16-s + 1.02e7·17-s − 1.27e7·19-s + 1.21e7·20-s + 2.32e7·22-s + 2.74e7·23-s + 9.23e7·25-s + 6.06e7·26-s + 1.72e7·28-s + 1.08e8·29-s − 2.02e8·31-s − 3.35e7·32-s − 3.27e8·34-s + 1.99e8·35-s + 4.12e8·37-s + 4.09e8·38-s − 3.89e8·40-s + 2.45e8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.70·5-s + 0.377·7-s − 0.353·8-s − 1.20·10-s − 1.36·11-s − 1.41·13-s − 0.267·14-s + 1/4·16-s + 1.74·17-s − 1.18·19-s + 0.850·20-s + 0.962·22-s + 0.888·23-s + 1.89·25-s + 1.00·26-s + 0.188·28-s + 0.978·29-s − 1.26·31-s − 0.176·32-s − 1.23·34-s + 0.642·35-s + 0.977·37-s + 0.838·38-s − 0.601·40-s + 0.330·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.080412470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.080412470\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{5} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 - 2376 p T + p^{11} T^{2} \) |
| 11 | \( 1 + 727110 T + p^{11} T^{2} \) |
| 13 | \( 1 + 1895734 T + p^{11} T^{2} \) |
| 17 | \( 1 - 10233912 T + p^{11} T^{2} \) |
| 19 | \( 1 + 12792796 T + p^{11} T^{2} \) |
| 23 | \( 1 - 27412290 T + p^{11} T^{2} \) |
| 29 | \( 1 - 108082914 T + p^{11} T^{2} \) |
| 31 | \( 1 + 202243384 T + p^{11} T^{2} \) |
| 37 | \( 1 - 412454954 T + p^{11} T^{2} \) |
| 41 | \( 1 - 245108604 T + p^{11} T^{2} \) |
| 43 | \( 1 - 509839844 T + p^{11} T^{2} \) |
| 47 | \( 1 + 699876996 T + p^{11} T^{2} \) |
| 53 | \( 1 - 4836690462 T + p^{11} T^{2} \) |
| 59 | \( 1 + 2462248272 T + p^{11} T^{2} \) |
| 61 | \( 1 - 9054716702 T + p^{11} T^{2} \) |
| 67 | \( 1 + 2923148584 T + p^{11} T^{2} \) |
| 71 | \( 1 - 11268869310 T + p^{11} T^{2} \) |
| 73 | \( 1 - 2809231382 T + p^{11} T^{2} \) |
| 79 | \( 1 - 25573574516 T + p^{11} T^{2} \) |
| 83 | \( 1 - 38718767364 T + p^{11} T^{2} \) |
| 89 | \( 1 + 90783707844 T + p^{11} T^{2} \) |
| 97 | \( 1 - 130559880038 T + p^{11} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76898763310252886758364026353, −10.12036415397397225917288846136, −9.401101520911872264986134151162, −8.142328122100769933614295449094, −7.10178777781330374444340824151, −5.75287266914271690895947298015, −5.04216690122801119099700455449, −2.74198835081147138546537204407, −2.06008302473848892122996785828, −0.77518676121257283639854757762,
0.77518676121257283639854757762, 2.06008302473848892122996785828, 2.74198835081147138546537204407, 5.04216690122801119099700455449, 5.75287266914271690895947298015, 7.10178777781330374444340824151, 8.142328122100769933614295449094, 9.401101520911872264986134151162, 10.12036415397397225917288846136, 10.76898763310252886758364026353
