| L(s) = 1 | − 2-s + 3·3-s − 3·6-s + 5·7-s + 8-s + 3·9-s − 4·13-s − 5·14-s − 16-s + 2·17-s − 3·18-s + 2·19-s + 15·21-s + 23-s + 3·24-s + 4·26-s − 2·29-s − 10·31-s − 2·34-s + 8·37-s − 2·38-s − 12·39-s − 6·41-s − 15·42-s + 10·43-s − 46-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1.22·6-s + 1.88·7-s + 0.353·8-s + 9-s − 1.10·13-s − 1.33·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 0.458·19-s + 3.27·21-s + 0.208·23-s + 0.612·24-s + 0.784·26-s − 0.371·29-s − 1.79·31-s − 0.342·34-s + 1.31·37-s − 0.324·38-s − 1.92·39-s − 0.937·41-s − 2.31·42-s + 1.52·43-s − 0.147·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.229833526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.229833526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.40069568547317101548102551534, −11.39672349794215585897234203064, −10.70579498926116753181838898832, −10.20388157130398291704756193339, −9.676889340828289790044672201961, −9.134326316220532963709506001841, −9.072981499623146128113275548746, −8.238153075996555691246610215813, −8.219878552554470314574620495250, −7.56765317661854420775095178722, −7.49981925385182309215577814242, −6.79608033597640837649799375189, −5.74640849959557748866390971630, −5.13347973556507337405944809717, −4.84614784415942484536992430751, −3.93668841716346014200728726322, −3.49151910112258463507541597741, −2.37431024570718377028093746445, −2.25996547662166844399380393455, −1.20332184231598067930233316127,
1.20332184231598067930233316127, 2.25996547662166844399380393455, 2.37431024570718377028093746445, 3.49151910112258463507541597741, 3.93668841716346014200728726322, 4.84614784415942484536992430751, 5.13347973556507337405944809717, 5.74640849959557748866390971630, 6.79608033597640837649799375189, 7.49981925385182309215577814242, 7.56765317661854420775095178722, 8.219878552554470314574620495250, 8.238153075996555691246610215813, 9.072981499623146128113275548746, 9.134326316220532963709506001841, 9.676889340828289790044672201961, 10.20388157130398291704756193339, 10.70579498926116753181838898832, 11.39672349794215585897234203064, 11.40069568547317101548102551534
