| L(s) = 1 | − 2-s + 4-s + 2·5-s + 2·7-s − 8-s − 2·10-s − 2·14-s + 16-s − 17-s − 6·19-s + 2·20-s + 6·23-s − 25-s + 2·28-s − 2·29-s + 4·31-s − 32-s + 34-s + 4·35-s − 10·37-s + 6·38-s − 2·40-s − 6·41-s + 10·43-s − 6·46-s + 12·47-s − 3·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.755·7-s − 0.353·8-s − 0.632·10-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 1.37·19-s + 0.447·20-s + 1.25·23-s − 1/5·25-s + 0.377·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.171·34-s + 0.676·35-s − 1.64·37-s + 0.973·38-s − 0.316·40-s − 0.937·41-s + 1.52·43-s − 0.884·46-s + 1.75·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−15.10671194715399, −14.75262889486528, −14.08088650846727, −13.60410067263616, −13.11341037305934, −12.44359511858478, −11.98305283112523, −11.22746047852571, −10.89762213219379, −10.23602857189852, −10.03431728589687, −9.009730933160014, −8.863183486131093, −8.394882777664065, −7.496699807253265, −7.158286516325917, −6.442101048880457, −5.875033241858735, −5.347198648794559, −4.612955629150540, −4.009363912082310, −3.009307518545189, −2.341534980199933, −1.772859187232849, −1.121746731840023, 0,
1.121746731840023, 1.772859187232849, 2.341534980199933, 3.009307518545189, 4.009363912082310, 4.612955629150540, 5.347198648794559, 5.875033241858735, 6.442101048880457, 7.158286516325917, 7.496699807253265, 8.394882777664065, 8.863183486131093, 9.009730933160014, 10.03431728589687, 10.23602857189852, 10.89762213219379, 11.22746047852571, 11.98305283112523, 12.44359511858478, 13.11341037305934, 13.60410067263616, 14.08088650846727, 14.75262889486528, 15.10671194715399
