| L(s) = 1 | + 243·3-s + 2.86e3·5-s + 9.12e3·7-s + 5.90e4·9-s + 6.68e5·11-s + 2.05e6·13-s + 6.95e5·15-s + 1.60e6·17-s − 2.30e5·19-s + 2.21e6·21-s − 4.30e7·23-s − 4.06e7·25-s + 1.43e7·27-s − 1.41e8·29-s + 2.33e8·31-s + 1.62e8·33-s + 2.61e7·35-s + 2.78e8·37-s + 4.98e8·39-s − 1.18e9·41-s + 8.56e8·43-s + 1.68e8·45-s − 1.66e9·47-s − 1.89e9·49-s + 3.89e8·51-s − 3.85e9·53-s + 1.91e9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.409·5-s + 0.205·7-s + 1/3·9-s + 1.25·11-s + 1.53·13-s + 0.236·15-s + 0.274·17-s − 0.0213·19-s + 0.118·21-s − 1.39·23-s − 0.832·25-s + 0.192·27-s − 1.28·29-s + 1.46·31-s + 0.722·33-s + 0.0840·35-s + 0.659·37-s + 0.885·39-s − 1.59·41-s + 0.888·43-s + 0.136·45-s − 1.05·47-s − 0.957·49-s + 0.158·51-s − 1.26·53-s + 0.512·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.381044224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.381044224\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 - 2862 T + p^{11} T^{2} \) |
| 7 | \( 1 - 1304 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 668196 T + p^{11} T^{2} \) |
| 13 | \( 1 - 2052950 T + p^{11} T^{2} \) |
| 17 | \( 1 - 1604178 T + p^{11} T^{2} \) |
| 19 | \( 1 + 230500 T + p^{11} T^{2} \) |
| 23 | \( 1 + 43012728 T + p^{11} T^{2} \) |
| 29 | \( 1 + 141745194 T + p^{11} T^{2} \) |
| 31 | \( 1 - 233221904 T + p^{11} T^{2} \) |
| 37 | \( 1 - 278269694 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1181577510 T + p^{11} T^{2} \) |
| 43 | \( 1 - 856975172 T + p^{11} T^{2} \) |
| 47 | \( 1 + 35405424 p T + p^{11} T^{2} \) |
| 53 | \( 1 + 3851181666 T + p^{11} T^{2} \) |
| 59 | \( 1 - 10339000596 T + p^{11} T^{2} \) |
| 61 | \( 1 - 185948102 T + p^{11} T^{2} \) |
| 67 | \( 1 - 2915010572 T + p^{11} T^{2} \) |
| 71 | \( 1 - 12662314200 T + p^{11} T^{2} \) |
| 73 | \( 1 + 15201270694 T + p^{11} T^{2} \) |
| 79 | \( 1 + 36644027488 T + p^{11} T^{2} \) |
| 83 | \( 1 + 9217637028 T + p^{11} T^{2} \) |
| 89 | \( 1 - 30573828810 T + p^{11} T^{2} \) |
| 97 | \( 1 - 145701815906 T + p^{11} T^{2} \) |
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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
−17.51791686704730959012293030337, −15.98925637469196740731447052831, −14.46054150958120971931926396400, −13.40486673938297547002512583911, −11.57685989951530715570327031007, −9.730706223453636891418840058371, −8.286740107669599435933241542089, −6.23386803445897243129913007942, −3.81563409850929373405982822853, −1.57446061939623492499543256256,
1.57446061939623492499543256256, 3.81563409850929373405982822853, 6.23386803445897243129913007942, 8.286740107669599435933241542089, 9.730706223453636891418840058371, 11.57685989951530715570327031007, 13.40486673938297547002512583911, 14.46054150958120971931926396400, 15.98925637469196740731447052831, 17.51791686704730959012293030337
