| L(s) = 1 | − 30·3-s − 32·5-s + 49·7-s + 657·9-s + 624·11-s + 708·13-s + 960·15-s + 934·17-s − 1.85e3·19-s − 1.47e3·21-s − 1.12e3·23-s − 2.10e3·25-s − 1.24e4·27-s + 1.17e3·29-s + 2.90e3·31-s − 1.87e4·33-s − 1.56e3·35-s + 1.24e4·37-s − 2.12e4·39-s + 2.66e3·41-s + 7.14e3·43-s − 2.10e4·45-s − 7.46e3·47-s + 2.40e3·49-s − 2.80e4·51-s + 2.72e4·53-s − 1.99e4·55-s + ⋯ |
| L(s) = 1 | − 1.92·3-s − 0.572·5-s + 0.377·7-s + 2.70·9-s + 1.55·11-s + 1.16·13-s + 1.10·15-s + 0.783·17-s − 1.18·19-s − 0.727·21-s − 0.441·23-s − 0.672·25-s − 3.27·27-s + 0.259·29-s + 0.543·31-s − 2.99·33-s − 0.216·35-s + 1.49·37-s − 2.23·39-s + 0.247·41-s + 0.589·43-s − 1.54·45-s − 0.493·47-s + 1/7·49-s − 1.50·51-s + 1.33·53-s − 0.890·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.148257291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.148257291\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 10 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 32 T + p^{5} T^{2} \) |
| 11 | \( 1 - 624 T + p^{5} T^{2} \) |
| 13 | \( 1 - 708 T + p^{5} T^{2} \) |
| 17 | \( 1 - 934 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1858 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1120 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1174 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2908 T + p^{5} T^{2} \) |
| 37 | \( 1 - 12462 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2662 T + p^{5} T^{2} \) |
| 43 | \( 1 - 7144 T + p^{5} T^{2} \) |
| 47 | \( 1 + 7468 T + p^{5} T^{2} \) |
| 53 | \( 1 - 27274 T + p^{5} T^{2} \) |
| 59 | \( 1 + 2490 T + p^{5} T^{2} \) |
| 61 | \( 1 - 11096 T + p^{5} T^{2} \) |
| 67 | \( 1 + 39756 T + p^{5} T^{2} \) |
| 71 | \( 1 + 69888 T + p^{5} T^{2} \) |
| 73 | \( 1 - 16450 T + p^{5} T^{2} \) |
| 79 | \( 1 - 78376 T + p^{5} T^{2} \) |
| 83 | \( 1 + 109818 T + p^{5} T^{2} \) |
| 89 | \( 1 + 56966 T + p^{5} T^{2} \) |
| 97 | \( 1 + 115946 T + p^{5} T^{2} \) |
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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
−10.59976398878111244350784911711, −9.659592409662915758039925554300, −8.395857463240365325373463821657, −7.26631235370894855647947059131, −6.27099710758987353035196691321, −5.84346409180883066961602031067, −4.41249820053251753809863227449, −3.95251649833152450660540702166, −1.50830869087911875995077581111, −0.67954115003592206253094517732,
0.67954115003592206253094517732, 1.50830869087911875995077581111, 3.95251649833152450660540702166, 4.41249820053251753809863227449, 5.84346409180883066961602031067, 6.27099710758987353035196691321, 7.26631235370894855647947059131, 8.395857463240365325373463821657, 9.659592409662915758039925554300, 10.59976398878111244350784911711
