| 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
| \(L(3)\) |
\(\approx\) |
\(4.187590150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.187590150\) |
| \(L(\frac{7}{2})\) |
|
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
−9.996118724105065156668466334681, −9.345043319960076057262833342846, −8.230570626432969136775843943954, −7.900777959191941085034757036604, −7.03178155852422763538641808158, −5.43543856737362525337529338711, −4.00768697279557450238735317728, −3.27488949204659806931351649637, −2.45631589271709401571224111849, −1.00755590478842028016131675157,
1.00755590478842028016131675157, 2.45631589271709401571224111849, 3.27488949204659806931351649637, 4.00768697279557450238735317728, 5.43543856737362525337529338711, 7.03178155852422763538641808158, 7.900777959191941085034757036604, 8.230570626432969136775843943954, 9.345043319960076057262833342846, 9.996118724105065156668466334681
