| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 19·5-s + 6·6-s + 9·7-s − 8·8-s + 9·9-s + 38·10-s − 13·11-s − 12·12-s + 38·13-s − 18·14-s + 57·15-s + 16·16-s + 99·17-s − 18·18-s − 19·19-s − 76·20-s − 27·21-s + 26·22-s + 68·23-s + 24·24-s + 236·25-s − 76·26-s − 27·27-s + 36·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.69·5-s + 0.408·6-s + 0.485·7-s − 0.353·8-s + 1/3·9-s + 1.20·10-s − 0.356·11-s − 0.288·12-s + 0.810·13-s − 0.343·14-s + 0.981·15-s + 1/4·16-s + 1.41·17-s − 0.235·18-s − 0.229·19-s − 0.849·20-s − 0.280·21-s + 0.251·22-s + 0.616·23-s + 0.204·24-s + 1.88·25-s − 0.573·26-s − 0.192·27-s + 0.242·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7029246485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7029246485\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 19 | \( 1 + p T \) |
| good | 5 | \( 1 + 19 T + p^{3} T^{2} \) |
| 7 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 + 13 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 99 T + p^{3} T^{2} \) |
| 23 | \( 1 - 68 T + p^{3} T^{2} \) |
| 29 | \( 1 - 130 T + p^{3} T^{2} \) |
| 31 | \( 1 - 262 T + p^{3} T^{2} \) |
| 37 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 8 T + p^{3} T^{2} \) |
| 43 | \( 1 - 73 T + p^{3} T^{2} \) |
| 47 | \( 1 + 271 T + p^{3} T^{2} \) |
| 53 | \( 1 + 502 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 587 T + p^{3} T^{2} \) |
| 67 | \( 1 - 684 T + p^{3} T^{2} \) |
| 71 | \( 1 - 992 T + p^{3} T^{2} \) |
| 73 | \( 1 + 507 T + p^{3} T^{2} \) |
| 79 | \( 1 - 980 T + p^{3} T^{2} \) |
| 83 | \( 1 + 492 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1046 T + p^{3} T^{2} \) |
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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
−12.66131547595628992419080935466, −11.77627518195393443086476826268, −11.14084472826655380607764445591, −10.14220787723011666683002373846, −8.439917205452356167492388358882, −7.86363925676123593651742789728, −6.66805317240232257149374681429, −4.96984742641435997064892642017, −3.46433789105938249651733016152, −0.850172726347256098789351208515,
0.850172726347256098789351208515, 3.46433789105938249651733016152, 4.96984742641435997064892642017, 6.66805317240232257149374681429, 7.86363925676123593651742789728, 8.439917205452356167492388358882, 10.14220787723011666683002373846, 11.14084472826655380607764445591, 11.77627518195393443086476826268, 12.66131547595628992419080935466
