| L(s) = 1 | − 2.70e3·3-s − 2.51e5·5-s − 1.37e6·7-s − 7.05e6·9-s + 4.32e7·11-s − 3.23e8·13-s + 6.80e8·15-s − 1.91e8·17-s + 6.51e9·19-s + 3.70e9·21-s − 2.38e10·23-s + 3.29e10·25-s + 5.78e10·27-s + 1.76e11·29-s + 1.52e11·31-s − 1.16e11·33-s + 3.46e11·35-s + 2.15e10·37-s + 8.72e11·39-s − 2.45e11·41-s − 2.76e12·43-s + 1.77e12·45-s − 2.81e12·47-s − 2.85e12·49-s + 5.17e11·51-s − 3.49e12·53-s − 1.09e13·55-s + ⋯ |
| L(s) = 1 | − 0.712·3-s − 1.44·5-s − 0.630·7-s − 0.491·9-s + 0.669·11-s − 1.42·13-s + 1.02·15-s − 0.113·17-s + 1.67·19-s + 0.449·21-s − 1.46·23-s + 1.07·25-s + 1.06·27-s + 1.90·29-s + 0.992·31-s − 0.477·33-s + 0.909·35-s + 0.0373·37-s + 1.01·39-s − 0.196·41-s − 1.55·43-s + 0.709·45-s − 0.809·47-s − 0.602·49-s + 0.0807·51-s − 0.408·53-s − 0.965·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.5757125299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5757125299\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 100 p^{3} T + p^{15} T^{2} \) |
| 5 | \( 1 + 50378 p T + p^{15} T^{2} \) |
| 7 | \( 1 + 196296 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 3935156 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 323161466 T + p^{15} T^{2} \) |
| 17 | \( 1 + 191653646 T + p^{15} T^{2} \) |
| 19 | \( 1 - 6515456644 T + p^{15} T^{2} \) |
| 23 | \( 1 + 23880801512 T + p^{15} T^{2} \) |
| 29 | \( 1 - 176820596982 T + p^{15} T^{2} \) |
| 31 | \( 1 - 152007193888 T + p^{15} T^{2} \) |
| 37 | \( 1 - 21581233902 T + p^{15} T^{2} \) |
| 41 | \( 1 + 245334499686 T + p^{15} T^{2} \) |
| 43 | \( 1 + 2769961534756 T + p^{15} T^{2} \) |
| 47 | \( 1 + 2811771943248 T + p^{15} T^{2} \) |
| 53 | \( 1 + 3491413730722 T + p^{15} T^{2} \) |
| 59 | \( 1 - 15827800893676 T + p^{15} T^{2} \) |
| 61 | \( 1 + 24609047974442 T + p^{15} T^{2} \) |
| 67 | \( 1 - 20706233653684 T + p^{15} T^{2} \) |
| 71 | \( 1 - 719982528200 T + p^{15} T^{2} \) |
| 73 | \( 1 - 29883036220282 T + p^{15} T^{2} \) |
| 79 | \( 1 - 148100908648400 T + p^{15} T^{2} \) |
| 83 | \( 1 - 302806756982468 T + p^{15} T^{2} \) |
| 89 | \( 1 + 496150966996374 T + p^{15} T^{2} \) |
| 97 | \( 1 - 309183128990882 T + p^{15} T^{2} \) |
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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
−15.68099836541238319849099900752, −14.20411759121809858745977816082, −12.04495418091340258624350538203, −11.77653406063503479207182819724, −9.919655217223083970412042272574, −8.050264802497350460682673477249, −6.61395658344741064307662256259, −4.79779126188904728812577126694, −3.18403779256160235614301357741, −0.50637971763188014959486836674,
0.50637971763188014959486836674, 3.18403779256160235614301357741, 4.79779126188904728812577126694, 6.61395658344741064307662256259, 8.050264802497350460682673477249, 9.919655217223083970412042272574, 11.77653406063503479207182819724, 12.04495418091340258624350538203, 14.20411759121809858745977816082, 15.68099836541238319849099900752
