| L(s) = 1 | + 2·5-s − 7-s + 11-s − 13-s − 2·17-s − 2·19-s − 5·23-s + 3·25-s − 29-s − 5·31-s − 2·35-s + 12·37-s + 12·41-s − 2·43-s + 13·47-s − 5·49-s + 3·53-s + 2·55-s − 10·59-s + 14·61-s − 2·65-s − 10·67-s + 3·71-s + 18·73-s − 77-s − 6·79-s + 20·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.301·11-s − 0.277·13-s − 0.485·17-s − 0.458·19-s − 1.04·23-s + 3/5·25-s − 0.185·29-s − 0.898·31-s − 0.338·35-s + 1.97·37-s + 1.87·41-s − 0.304·43-s + 1.89·47-s − 5/7·49-s + 0.412·53-s + 0.269·55-s − 1.30·59-s + 1.79·61-s − 0.248·65-s − 1.22·67-s + 0.356·71-s + 2.10·73-s − 0.113·77-s − 0.675·79-s + 2.19·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.726965735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.726965735\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 50 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 13 T + 128 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 136 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 134 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.962043025864982669306749056818, −7.910771313327960846788455889623, −7.50569686030900290527287329541, −7.17056666702544373306438727644, −6.64974847630523149292415149209, −6.35276403401918747030593594470, −6.07215948562587193459655444334, −5.91231305282293505736667727669, −5.29281475005438318266270051986, −5.14321634669607670616038103209, −4.42373336766855541428134190934, −4.33540624089438248348651091042, −3.67063152579970817389143199370, −3.64272337362877636234521341025, −2.72801610236804218385578608138, −2.44869037646145791190438075389, −2.25000843542512515779387954030, −1.63753256812706471389335105593, −1.02626896254072784885535362974, −0.44762648697954252549713153310,
0.44762648697954252549713153310, 1.02626896254072784885535362974, 1.63753256812706471389335105593, 2.25000843542512515779387954030, 2.44869037646145791190438075389, 2.72801610236804218385578608138, 3.64272337362877636234521341025, 3.67063152579970817389143199370, 4.33540624089438248348651091042, 4.42373336766855541428134190934, 5.14321634669607670616038103209, 5.29281475005438318266270051986, 5.91231305282293505736667727669, 6.07215948562587193459655444334, 6.35276403401918747030593594470, 6.64974847630523149292415149209, 7.17056666702544373306438727644, 7.50569686030900290527287329541, 7.910771313327960846788455889623, 7.962043025864982669306749056818
