| L(s) = 1 | − 46·3-s − 625·5-s + 1.03e4·7-s − 1.75e4·9-s + 5.56e3·11-s + 4.59e4·13-s + 2.87e4·15-s − 3.81e5·17-s − 6.10e5·19-s − 4.74e5·21-s + 1.44e6·23-s + 3.90e5·25-s + 1.71e6·27-s + 5.38e6·29-s − 3.05e6·31-s − 2.56e5·33-s − 6.44e6·35-s + 1.28e7·37-s − 2.11e6·39-s − 3.37e7·41-s + 3.68e7·43-s + 1.09e7·45-s + 4.41e7·47-s + 6.61e7·49-s + 1.75e7·51-s + 2.97e7·53-s − 3.48e6·55-s + ⋯ |
| L(s) = 1 | − 0.327·3-s − 0.447·5-s + 1.62·7-s − 0.892·9-s + 0.114·11-s + 0.446·13-s + 0.146·15-s − 1.10·17-s − 1.07·19-s − 0.532·21-s + 1.07·23-s + 1/5·25-s + 0.620·27-s + 1.41·29-s − 0.593·31-s − 0.0375·33-s − 0.726·35-s + 1.13·37-s − 0.146·39-s − 1.86·41-s + 1.64·43-s + 0.399·45-s + 1.32·47-s + 1.63·49-s + 0.363·51-s + 0.517·53-s − 0.0512·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.784258029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.784258029\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{4} T \) |
| good | 3 | \( 1 + 46 T + p^{9} T^{2} \) |
| 7 | \( 1 - 1474 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 5568 T + p^{9} T^{2} \) |
| 13 | \( 1 - 45986 T + p^{9} T^{2} \) |
| 17 | \( 1 + 381318 T + p^{9} T^{2} \) |
| 19 | \( 1 + 610460 T + p^{9} T^{2} \) |
| 23 | \( 1 - 1447914 T + p^{9} T^{2} \) |
| 29 | \( 1 - 5385510 T + p^{9} T^{2} \) |
| 31 | \( 1 + 3053852 T + p^{9} T^{2} \) |
| 37 | \( 1 - 12889442 T + p^{9} T^{2} \) |
| 41 | \( 1 + 33786618 T + p^{9} T^{2} \) |
| 43 | \( 1 - 36886234 T + p^{9} T^{2} \) |
| 47 | \( 1 - 44163798 T + p^{9} T^{2} \) |
| 53 | \( 1 - 29746266 T + p^{9} T^{2} \) |
| 59 | \( 1 - 65575380 T + p^{9} T^{2} \) |
| 61 | \( 1 - 40183202 T + p^{9} T^{2} \) |
| 67 | \( 1 - 115706158 T + p^{9} T^{2} \) |
| 71 | \( 1 - 231681708 T + p^{9} T^{2} \) |
| 73 | \( 1 - 358691906 T + p^{9} T^{2} \) |
| 79 | \( 1 - 486017080 T + p^{9} T^{2} \) |
| 83 | \( 1 + 251168886 T + p^{9} T^{2} \) |
| 89 | \( 1 + 526039110 T + p^{9} T^{2} \) |
| 97 | \( 1 + 1075981438 T + p^{9} T^{2} \) |
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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
−12.30298920822210325403700126375, −11.19266051199399367811229657667, −10.87993718169118502101559896324, −8.808745316593007180669582585912, −8.182888086151598451083468973623, −6.70459058739100202788348149578, −5.26127999052692073932817683791, −4.22184330745030538410008162234, −2.36263258061719231346246709414, −0.807935538238711261654465584590,
0.807935538238711261654465584590, 2.36263258061719231346246709414, 4.22184330745030538410008162234, 5.26127999052692073932817683791, 6.70459058739100202788348149578, 8.182888086151598451083468973623, 8.808745316593007180669582585912, 10.87993718169118502101559896324, 11.19266051199399367811229657667, 12.30298920822210325403700126375
