| L(s) = 1 | − 2-s − 3-s + 6-s + 3·7-s + 8-s + 3·11-s + 5·13-s − 3·14-s − 16-s − 3·19-s − 3·21-s − 3·22-s − 4·23-s − 24-s − 5·26-s + 27-s + 4·29-s + 12·31-s − 3·33-s + 9·37-s + 3·38-s − 5·39-s + 10·41-s + 3·42-s − 10·43-s + 4·46-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 0.904·11-s + 1.38·13-s − 0.801·14-s − 1/4·16-s − 0.688·19-s − 0.654·21-s − 0.639·22-s − 0.834·23-s − 0.204·24-s − 0.980·26-s + 0.192·27-s + 0.742·29-s + 2.15·31-s − 0.522·33-s + 1.47·37-s + 0.486·38-s − 0.800·39-s + 1.56·41-s + 0.462·42-s − 1.52·43-s + 0.589·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.866037989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.866037989\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 14 T + 125 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.220679743937605790105560463887, −9.060429347822173822436809890929, −8.527084788956893696958218148730, −8.145359499102354543690118284547, −7.914192362404614805719572018465, −7.82623802703187335432621076859, −6.79264625458891936640621092117, −6.69120006159615784795811198113, −6.27381733532605385172864484989, −5.81957153357429168895677594575, −5.60031059973353453200619198592, −4.60824271378925341149215040773, −4.50752833785881212612706935171, −4.38203075637276440088742411357, −3.53580232822593334805412243621, −3.08653212505917891754161351994, −2.25969933927803109755320388676, −1.77056853878608768810911377284, −1.09027320388650284101249275930, −0.72436030524830028311236538614,
0.72436030524830028311236538614, 1.09027320388650284101249275930, 1.77056853878608768810911377284, 2.25969933927803109755320388676, 3.08653212505917891754161351994, 3.53580232822593334805412243621, 4.38203075637276440088742411357, 4.50752833785881212612706935171, 4.60824271378925341149215040773, 5.60031059973353453200619198592, 5.81957153357429168895677594575, 6.27381733532605385172864484989, 6.69120006159615784795811198113, 6.79264625458891936640621092117, 7.82623802703187335432621076859, 7.914192362404614805719572018465, 8.145359499102354543690118284547, 8.527084788956893696958218148730, 9.060429347822173822436809890929, 9.220679743937605790105560463887
