| L(s) = 1 | − 7-s + 3·11-s + 4·13-s + 12·17-s − 8·19-s + 5·25-s − 6·29-s − 2·31-s − 14·37-s + 12·41-s + 7·43-s + 6·47-s + 6·53-s + 6·59-s − 2·61-s + 13·67-s + 18·71-s + 16·73-s − 3·77-s + 79-s + 24·89-s − 4·91-s + 4·97-s + 12·101-s + 4·103-s + 30·107-s − 20·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.904·11-s + 1.10·13-s + 2.91·17-s − 1.83·19-s + 25-s − 1.11·29-s − 0.359·31-s − 2.30·37-s + 1.87·41-s + 1.06·43-s + 0.875·47-s + 0.824·53-s + 0.781·59-s − 0.256·61-s + 1.58·67-s + 2.13·71-s + 1.87·73-s − 0.341·77-s + 0.112·79-s + 2.54·89-s − 0.419·91-s + 0.406·97-s + 1.19·101-s + 0.394·103-s + 2.90·107-s − 1.91·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.394476220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.394476220\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 4 T - 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.093308475365437968465546386558, −8.839900672842803972448114318297, −8.572109849616381661966480774781, −8.065482307080618189075127903808, −7.52817366989395515514918279869, −7.50776805415091959310663454863, −6.67622163062769627203110577416, −6.62937625420447190797270617350, −6.00347801615892445842681025030, −5.81867265277218603701357459443, −5.19778049305975904981262210196, −5.07732058607444408176928771367, −4.05711625852346164691042724454, −3.95973663972225171632125177140, −3.39516022945208565971005374601, −3.33734822986013177494008665221, −2.22251729519133139387045464012, −2.05676049221387197329680689116, −1.00004717034460877358704139867, −0.832186807191607495774841564140,
0.832186807191607495774841564140, 1.00004717034460877358704139867, 2.05676049221387197329680689116, 2.22251729519133139387045464012, 3.33734822986013177494008665221, 3.39516022945208565971005374601, 3.95973663972225171632125177140, 4.05711625852346164691042724454, 5.07732058607444408176928771367, 5.19778049305975904981262210196, 5.81867265277218603701357459443, 6.00347801615892445842681025030, 6.62937625420447190797270617350, 6.67622163062769627203110577416, 7.50776805415091959310663454863, 7.52817366989395515514918279869, 8.065482307080618189075127903808, 8.572109849616381661966480774781, 8.839900672842803972448114318297, 9.093308475365437968465546386558
