| L(s) = 1 | + 2-s + 3·3-s + 4-s + 2·5-s + 3·6-s + 8-s + 6·9-s + 2·10-s − 11-s + 3·12-s − 7·13-s + 6·15-s + 16-s + 2·17-s + 6·18-s + 2·20-s − 22-s − 8·23-s + 3·24-s − 25-s − 7·26-s + 9·27-s − 5·29-s + 6·30-s + 4·31-s + 32-s − 3·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.894·5-s + 1.22·6-s + 0.353·8-s + 2·9-s + 0.632·10-s − 0.301·11-s + 0.866·12-s − 1.94·13-s + 1.54·15-s + 1/4·16-s + 0.485·17-s + 1.41·18-s + 0.447·20-s − 0.213·22-s − 1.66·23-s + 0.612·24-s − 1/5·25-s − 1.37·26-s + 1.73·27-s − 0.928·29-s + 1.09·30-s + 0.718·31-s + 0.176·32-s − 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.685652829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.685652829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−9.856353077622187691368346097215, −9.232984660708510751075852428852, −7.949688248994453332985143052492, −7.66334045549742507331716148152, −6.56153010353145194477530352450, −5.46355028016127947092103451217, −4.50301199161709359138099430647, −3.48832092996576829359255750084, −2.43990333774251370043130333163, −2.00011124513055881492662257488,
2.00011124513055881492662257488, 2.43990333774251370043130333163, 3.48832092996576829359255750084, 4.50301199161709359138099430647, 5.46355028016127947092103451217, 6.56153010353145194477530352450, 7.66334045549742507331716148152, 7.949688248994453332985143052492, 9.232984660708510751075852428852, 9.856353077622187691368346097215
