L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s + 2·11-s + 12-s + 2·13-s + 2·15-s + 16-s + 6·17-s − 18-s − 4·19-s + 2·20-s − 2·22-s + 23-s − 24-s − 25-s − 2·26-s + 27-s + 2·29-s − 2·30-s − 2·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.603·11-s + 0.288·12-s + 0.554·13-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s − 0.426·22-s + 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.548366232\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.548366232\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.068858867900873410326289824372, −7.43095970267970765421658431374, −6.60673058010488749182250113226, −6.01424603921676345305833274594, −5.36000106720335748962426451738, −4.19888291598668511061960251508, −3.44867374142256507687550141420, −2.53848349245967536001807046270, −1.74680491888648407092379480010, −0.941709179073099274500484413897,
0.941709179073099274500484413897, 1.74680491888648407092379480010, 2.53848349245967536001807046270, 3.44867374142256507687550141420, 4.19888291598668511061960251508, 5.36000106720335748962426451738, 6.01424603921676345305833274594, 6.60673058010488749182250113226, 7.43095970267970765421658431374, 8.068858867900873410326289824372