L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s + 2·7-s + 8-s + 9-s + 3·10-s − 11-s − 12-s − 3·13-s + 2·14-s − 3·15-s + 16-s + 18-s − 4·19-s + 3·20-s − 2·21-s − 22-s − 4·23-s − 24-s + 4·25-s − 3·26-s − 27-s + 2·28-s + 7·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s − 0.832·13-s + 0.534·14-s − 0.774·15-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.670·20-s − 0.436·21-s − 0.213·22-s − 0.834·23-s − 0.204·24-s + 4/5·25-s − 0.588·26-s − 0.192·27-s + 0.377·28-s + 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.517770136\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.517770136\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−8.278856011624510743686879678650, −7.63855312229210610867391452956, −6.64395536676820069199652898928, −6.07535886062081717690928087339, −5.54165736205628367684540488757, −4.66754544926088490985162481927, −4.27809915441763530407269186017, −2.63139171119991095402867550204, −2.22254637564311131564552355629, −1.03814098671731685699204105431,
1.03814098671731685699204105431, 2.22254637564311131564552355629, 2.63139171119991095402867550204, 4.27809915441763530407269186017, 4.66754544926088490985162481927, 5.54165736205628367684540488757, 6.07535886062081717690928087339, 6.64395536676820069199652898928, 7.63855312229210610867391452956, 8.278856011624510743686879678650