| L(s) = 1 | − 32·2-s + 318·3-s + 1.02e3·4-s − 1.01e4·6-s + 7.07e4·7-s − 3.27e4·8-s − 7.60e4·9-s + 2.38e5·11-s + 3.25e5·12-s + 2.09e6·13-s − 2.26e6·14-s + 1.04e6·16-s − 5.95e6·17-s + 2.43e6·18-s + 1.02e7·19-s + 2.24e7·21-s − 7.62e6·22-s + 3.53e6·23-s − 1.04e7·24-s − 6.71e7·26-s − 8.05e7·27-s + 7.24e7·28-s − 1.39e8·29-s − 1.01e8·31-s − 3.35e7·32-s + 7.57e7·33-s + 1.90e8·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.755·3-s + 1/2·4-s − 0.534·6-s + 1.59·7-s − 0.353·8-s − 0.429·9-s + 0.446·11-s + 0.377·12-s + 1.56·13-s − 1.12·14-s + 1/4·16-s − 1.01·17-s + 0.303·18-s + 0.946·19-s + 1.20·21-s − 0.315·22-s + 0.114·23-s − 0.267·24-s − 1.10·26-s − 1.07·27-s + 0.795·28-s − 1.26·29-s − 0.633·31-s − 0.176·32-s + 0.337·33-s + 0.719·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.428237034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.428237034\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{5} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 106 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 10102 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 238272 T + p^{11} T^{2} \) |
| 13 | \( 1 - 2097478 T + p^{11} T^{2} \) |
| 17 | \( 1 + 5955546 T + p^{11} T^{2} \) |
| 19 | \( 1 - 10210820 T + p^{11} T^{2} \) |
| 23 | \( 1 - 3535758 T + p^{11} T^{2} \) |
| 29 | \( 1 + 139304850 T + p^{11} T^{2} \) |
| 31 | \( 1 + 101002348 T + p^{11} T^{2} \) |
| 37 | \( 1 - 524913814 T + p^{11} T^{2} \) |
| 41 | \( 1 - 284590422 T + p^{11} T^{2} \) |
| 43 | \( 1 - 1253635078 T + p^{11} T^{2} \) |
| 47 | \( 1 - 216106434 T + p^{11} T^{2} \) |
| 53 | \( 1 - 4881275358 T + p^{11} T^{2} \) |
| 59 | \( 1 - 8692473300 T + p^{11} T^{2} \) |
| 61 | \( 1 - 3296491802 T + p^{11} T^{2} \) |
| 67 | \( 1 + 18275027966 T + p^{11} T^{2} \) |
| 71 | \( 1 + 13287447588 T + p^{11} T^{2} \) |
| 73 | \( 1 - 32505250798 T + p^{11} T^{2} \) |
| 79 | \( 1 - 9297455960 T + p^{11} T^{2} \) |
| 83 | \( 1 - 22741484838 T + p^{11} T^{2} \) |
| 89 | \( 1 + 93378882390 T + p^{11} T^{2} \) |
| 97 | \( 1 - 5811134014 T + p^{11} T^{2} \) |
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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
−13.41589938842402765956275728417, −11.48419657542633205483041519437, −11.02318324518221891821339594658, −9.167487675812065729219710818994, −8.487846801499703004504771055731, −7.48393395186180922365708829364, −5.72625412209778471973601168790, −3.90149157416001918641463751125, −2.22972491462732227050849770578, −1.07875638258998647896368700764,
1.07875638258998647896368700764, 2.22972491462732227050849770578, 3.90149157416001918641463751125, 5.72625412209778471973601168790, 7.48393395186180922365708829364, 8.487846801499703004504771055731, 9.167487675812065729219710818994, 11.02318324518221891821339594658, 11.48419657542633205483041519437, 13.41589938842402765956275728417
