| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s − 3·9-s + 10-s − 4·11-s − 13-s − 14-s + 16-s + 3·18-s + 19-s − 20-s + 4·22-s − 4·23-s + 25-s + 26-s + 28-s − 5·29-s − 9·31-s − 32-s − 35-s − 3·36-s + 4·37-s − 38-s + 40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.707·18-s + 0.229·19-s − 0.223·20-s + 0.852·22-s − 0.834·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 0.928·29-s − 1.61·31-s − 0.176·32-s − 0.169·35-s − 1/2·36-s + 0.657·37-s − 0.162·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−10.80269217500459567842962810442, −9.789436282732918865000265521600, −8.782432063882567644596578764990, −7.971504104737887384111597076043, −7.36194765122952415589970596502, −5.95017742366424847842642438727, −5.04412930604684011969274249594, −3.43732947604446743308764448217, −2.17626482800391177771500986136, 0,
2.17626482800391177771500986136, 3.43732947604446743308764448217, 5.04412930604684011969274249594, 5.95017742366424847842642438727, 7.36194765122952415589970596502, 7.971504104737887384111597076043, 8.782432063882567644596578764990, 9.789436282732918865000265521600, 10.80269217500459567842962810442
