| L(s) = 1 | − 3-s − 2·7-s − 9-s − 11-s − 5·13-s + 5·17-s + 6·19-s + 2·21-s − 2·23-s + 29-s + 33-s − 12·37-s + 5·39-s + 2·41-s + 10·43-s − 5·47-s + 3·49-s − 5·51-s + 2·53-s − 6·57-s + 8·59-s + 6·61-s + 2·63-s + 4·67-s + 2·69-s − 16·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 1/3·9-s − 0.301·11-s − 1.38·13-s + 1.21·17-s + 1.37·19-s + 0.436·21-s − 0.417·23-s + 0.185·29-s + 0.174·33-s − 1.97·37-s + 0.800·39-s + 0.312·41-s + 1.52·43-s − 0.729·47-s + 3/7·49-s − 0.700·51-s + 0.274·53-s − 0.794·57-s + 1.04·59-s + 0.768·61-s + 0.251·63-s + 0.488·67-s + 0.240·69-s − 1.89·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.233131433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.233131433\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 174 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 108 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.903070344214789770762393679141, −8.726662338982278120377619147391, −8.211782136766845991644141940174, −7.67649774847495106267124097301, −7.35610656102963394479017205164, −7.31381384852572325405946113396, −6.70197260143630373282951468685, −6.30579390250905574959650460888, −5.68790731167795668622962690270, −5.67735465685009380091545602263, −5.12948443126867777100698169245, −4.92669009272901267617156952683, −4.29773236878074379164952736385, −3.73096639224400046604251600628, −3.16280650758089875663279859072, −3.14614415263466272735549677233, −2.30237679118507509764221342780, −1.95936067285970225251507194141, −0.962274490989772215868964614705, −0.45031073113195407174165662559,
0.45031073113195407174165662559, 0.962274490989772215868964614705, 1.95936067285970225251507194141, 2.30237679118507509764221342780, 3.14614415263466272735549677233, 3.16280650758089875663279859072, 3.73096639224400046604251600628, 4.29773236878074379164952736385, 4.92669009272901267617156952683, 5.12948443126867777100698169245, 5.67735465685009380091545602263, 5.68790731167795668622962690270, 6.30579390250905574959650460888, 6.70197260143630373282951468685, 7.31381384852572325405946113396, 7.35610656102963394479017205164, 7.67649774847495106267124097301, 8.211782136766845991644141940174, 8.726662338982278120377619147391, 8.903070344214789770762393679141
