| L(s) = 1 | + 2·2-s + 3·4-s − 3·5-s − 7-s + 4·8-s − 6·10-s + 4·13-s − 2·14-s + 5·16-s − 6·17-s + 4·19-s − 9·20-s − 6·23-s + 5·25-s + 8·26-s − 3·28-s + 3·29-s + 16·31-s + 6·32-s − 12·34-s + 3·35-s − 8·37-s + 8·38-s − 12·40-s + 6·41-s − 8·43-s − 12·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.34·5-s − 0.377·7-s + 1.41·8-s − 1.89·10-s + 1.10·13-s − 0.534·14-s + 5/4·16-s − 1.45·17-s + 0.917·19-s − 2.01·20-s − 1.25·23-s + 25-s + 1.56·26-s − 0.566·28-s + 0.557·29-s + 2.87·31-s + 1.06·32-s − 2.05·34-s + 0.507·35-s − 1.31·37-s + 1.29·38-s − 1.89·40-s + 0.937·41-s − 1.21·43-s − 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.743257487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.743257487\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.38886397933334988193916629202, −9.475663151672731156418072590876, −9.323973478791458326878873567121, −8.623408731786003754530177064383, −8.184663599171142558076462388587, −7.959001138170310168447772434151, −7.50967069718777235638547235447, −6.99933598987310551273753361842, −6.45375723586784867457889984428, −6.15101073263977076467663350953, −6.08746777451634720510656709676, −5.03705731211878736077582173957, −4.69758357275940886803603557316, −4.47542373088997820439236477953, −3.87182470730173416153711415545, −3.38537071491918194082525401274, −3.14216588243346435465642155812, −2.42667620499939067781944617911, −1.67467004161053159171230270879, −0.67872419458169656971759809767,
0.67872419458169656971759809767, 1.67467004161053159171230270879, 2.42667620499939067781944617911, 3.14216588243346435465642155812, 3.38537071491918194082525401274, 3.87182470730173416153711415545, 4.47542373088997820439236477953, 4.69758357275940886803603557316, 5.03705731211878736077582173957, 6.08746777451634720510656709676, 6.15101073263977076467663350953, 6.45375723586784867457889984428, 6.99933598987310551273753361842, 7.50967069718777235638547235447, 7.959001138170310168447772434151, 8.184663599171142558076462388587, 8.623408731786003754530177064383, 9.323973478791458326878873567121, 9.475663151672731156418072590876, 10.38886397933334988193916629202
