L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s − 11-s + 12-s − 4·13-s − 2·14-s + 16-s − 6·17-s − 18-s − 4·19-s + 2·21-s + 22-s + 6·23-s − 24-s − 5·25-s + 4·26-s + 27-s + 2·28-s + 6·29-s + 8·31-s − 32-s − 33-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.436·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7970914573\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7970914573\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.02818417573080606322340999692, −13.92129713751308299089397897707, −12.61238129145388540009030344017, −11.31231841896418545254169443203, −10.21971960261641202232640310225, −8.958855021658912251315559727512, −8.013104010237496671597514230187, −6.77506565559004391547494912024, −4.69333471332706230755932945396, −2.37545091357255938865951638025,
2.37545091357255938865951638025, 4.69333471332706230755932945396, 6.77506565559004391547494912024, 8.013104010237496671597514230187, 8.958855021658912251315559727512, 10.21971960261641202232640310225, 11.31231841896418545254169443203, 12.61238129145388540009030344017, 13.92129713751308299089397897707, 15.02818417573080606322340999692