| L(s) = 1 | − 2-s − 2·3-s + 4-s − 5-s + 2·6-s − 7-s − 8-s + 9-s + 10-s + 3·11-s − 2·12-s + 14-s + 2·15-s + 16-s − 6·17-s − 18-s + 5·19-s − 20-s + 2·21-s − 3·22-s + 2·24-s + 25-s + 4·27-s − 28-s − 2·30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s − 0.577·12-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.14·19-s − 0.223·20-s + 0.436·21-s − 0.639·22-s + 0.408·24-s + 1/5·25-s + 0.769·27-s − 0.188·28-s − 0.365·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.161546616794757596099393561421, −8.160273095798200994898651396861, −7.25611114446141217990928652783, −6.52596230714427548793513786310, −5.97158340388789448391905927019, −4.90557660459778148401551734970, −3.95651833930202986020544832493, −2.73515254909378110142167648146, −1.20588678809223336738925934160, 0,
1.20588678809223336738925934160, 2.73515254909378110142167648146, 3.95651833930202986020544832493, 4.90557660459778148401551734970, 5.97158340388789448391905927019, 6.52596230714427548793513786310, 7.25611114446141217990928652783, 8.160273095798200994898651396861, 9.161546616794757596099393561421
