L(s) = 1 | − 3-s − 3·4-s − 2·7-s + 9-s − 8·11-s + 3·12-s + 5·16-s + 12·17-s + 8·19-s + 2·21-s − 6·25-s − 27-s + 6·28-s + 4·29-s + 8·33-s − 3·36-s + 24·44-s − 5·48-s + 3·49-s − 12·51-s − 12·53-s − 8·57-s − 2·63-s − 3·64-s + 8·67-s − 36·68-s + 6·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 3/2·4-s − 0.755·7-s + 1/3·9-s − 2.41·11-s + 0.866·12-s + 5/4·16-s + 2.91·17-s + 1.83·19-s + 0.436·21-s − 6/5·25-s − 0.192·27-s + 1.13·28-s + 0.742·29-s + 1.39·33-s − 1/2·36-s + 3.61·44-s − 0.721·48-s + 3/7·49-s − 1.68·51-s − 1.64·53-s − 1.05·57-s − 0.251·63-s − 3/8·64-s + 0.977·67-s − 4.36·68-s + 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1271403 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1271403 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7573650337\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7573650337\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 31 | $C_2$ | \( 1 + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−7.81584315767456416244355813696, −7.81295430367747747098376616892, −7.43099978309389560690669930522, −6.60559600423068616268613662576, −5.94607665445287604157300008429, −5.72269195485958227685862562811, −5.20443893243519540369756938608, −5.02709416296405066539370067618, −4.69262259945753005310040105059, −3.59921807664512552463587180587, −3.49949852269448856369083761712, −3.05422074105458389777226041971, −2.22970928370065220133660210160, −1.05097518078079480859008704828, −0.50485220186860305883826488454,
0.50485220186860305883826488454, 1.05097518078079480859008704828, 2.22970928370065220133660210160, 3.05422074105458389777226041971, 3.49949852269448856369083761712, 3.59921807664512552463587180587, 4.69262259945753005310040105059, 5.02709416296405066539370067618, 5.20443893243519540369756938608, 5.72269195485958227685862562811, 5.94607665445287604157300008429, 6.60559600423068616268613662576, 7.43099978309389560690669930522, 7.81295430367747747098376616892, 7.81584315767456416244355813696